This content introduces fundamental concepts in number theory, focusing on divisibility rules, prime factorization, greatest common divisors (GCD), and least common multiples (LCM). It explains how to determine if a number is divisible by others using simple rules and how to break down numbers into their prime components.
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in this section we'll look at a couple
of topics from the field of number
Theory to start with number theory is
the study of the integers and the
properties that they possess it is one
of the oldest branches of mathematics
staying back to about 1800 BC
and while number three studies all
integers the positive and negative
integers in this lesson we'll focus on
just the positive integers so that is
we'll look at numbers like one two three
four five six and so on
now these are called the natural numbers
and we're going to let this letter N
represent the set of these numbers so to
do that I'm going to put these curly
brackets around these numbers and this
indicates that we're talking about a
collection a set of numbers and here the
script n is how we Define it so again
those are called the natural numbers
now what are some of the properties that
these numbers possess
when people are first examining these
numbers they took a number like let's
say six since it's on my list here and
they notice that oh well 6 is the same
thing as 2 plus 2 plus 2. you can split
it apart by using other numbers that
appear on this list and also you could
write this as three plus three
so the observation that was first made
was some of these numbers can be broken
apart with smaller numbers in fact we
could say 6 is equal to two times three
you can Factor it apart
but this isn't true for every single
number on the list like perhaps five if
you try to split five apart by using two
well it doesn't work it's no good so
sometimes you can break numbers apart
and other times you can't and this leads
us to our first definition
let A and B represent two positive
numbers we see that b is divisible by a
or that a divides B if there's a
remainder of zero when a is divided into B
B
so if this happens we have a notation
for it we write the following we write a
with this vertical line B and when you
see this this means
that a divides
divides B
B
and if a does not divide B we'll write
that like this a with a line with a
another line through it B and this means a
a
does not divide
divide B
B
so looking at the examples that we have
up here and this new notation that we
put down here
is it true that 2 divides 6. is this
true and the answer is yes
and if you actually check by dividing if
you try to divide 2 into 6 it goes in
three times and you're left with a
remainder of zero so this means two
divide six or six is divisible by 2.
likewise does 3 divide six does that
happen and the answer is also yes if you
take 3 and divide that into six it goes
in twice and you're left with a
remainder of zero
but does two divide five
no that does not happen so you can write
this 2 does not divide 5. and the reason
for this is if you take two and try to
divide it into five well the closest you
can get is four but this leaves you with
a remainder of one so if your remainder
is not zero it just means that two does
not divide this number
so these did divide into six here two
does not divide into five
with these examples in mind let's see if
we can come up with rules that tell us
when a number divides into another or not
in example one let's use long division
and a calculator to decide whether 2
divides into the following numbers 35
100 and let's see two trillion Thirty
billion four million five hundred fifty
five thousand six
so I'll start with 35 does 2 divide into
this number
now you might know the answer already
but if you're going to use long division
it would look like this you'd say well
let's take 2 divide it into 35 and see
if we get a remainder of 0 or not
well 2 goes into 3 Once
we're left over with 15.
to the closest you can get to 15 is by
multiplying by 7.
that gives you 14.
and so here we have a remainder of one
now what does that mean that means that
2 does not divide into 35.
you could also check this on a
calculator so I have a calculator here
let's see if you type in 35 divided by 2
so if you type in this divided by that
uh the value we get back is 17.5
and this means okay 2 did not evenly
divide into 35 and the reason for this
is well we have this decimal that's not
equal to zero after 17. so you can use
long division you'd even use a
calculator in either case 2 does not go
into 35.
now let's try this again with uh 100
so over here
we're going to divide 2 into 100 and see
what happens now again you might know
the answer already but let's try it with
long division
to divide 2 into 100 it first goes into
ten five times
and that worked out perfectly to give us
just zero left over and 2 goes into 0 0 times
times
and here we do in fact have a remainder
of zero and this means that 2 does in
fact divide 100 this is true
also obviously if you look at this in a
calculator if you type in 100 divided by
2 it tells you 50 which again is no
surprise but the fact that we didn't get
a decimal at the end means okay 2 does
evenly go into this number so this is true
true
all right and on to the last part of
this problem does two divide into this
large number here
you know this time I actually don't want
to use long division or a calculator the
reason is long division into this number
would take a long time and on a
calculator sometimes they don't even
have this many place values that you can
enter is there some other way to know if
this number is divisible by two or not
and the answer is yeah
if the number ends in an even number
itself then two will go into it so all
you really need to know is for this
number does it end in an even number if
it does then 2 will divide into it so
without doing any work we can actually
write the following statement and we
know it's true
2 divides into two trillion
30 billion
400 million five hundred fifty five
thousand six
this is a true statement we didn't have
to do any work like this or type
anything into a calculator I know it is
true because the number ends in an even number
number
if you know this fact well 50 ends in
even number so 2 must go into it but 35
ends in an odd number and so 2 was never
going to go into it
so this is known as a divisibility rule
for the number two if the number ends in
an even value you can divide 2 into it
otherwise you cannot and it saves you a
lot of work you don't have to do any of
this stuff
so that is a nice divisibility rule for
the number two but there's others for
numbers like 5 and 10 for example let's
talk about those now
if we go back to the number 35
does 5 go into 35 that is is this a true
statement does five evenly divide into 35
35
and the answer to this is yes
if you count by fives the number will
always end in five or zero so if you
look at just this last number because
it's 5 well I know five will evenly
divide into it
we could also ask this about does 5
divide into 100 is this going to be true
and the answer is also yes
again if you count by fives the number
either ends in a five or a zero so to
determine if this is true again you just
have to look at the last digit if it's
five or zero that means 5 will go into it
it
uh maybe I'll list out one more like
this what about the number 10 does 10
divide into 35
now the answer to this is no
and the reason is if you count by tens
you're going to skip over 35 you'll go
from 30 to 40.
does 10 divide into 100 and the answer
to this is yes
and that's because the number ends in a zero
zero
so for 2 5 and 10 you can determine
whether these numbers divide evenly into
a given value just by looking at the
last digit of a number
I will state these divisibility rules on
the next slide so we'll have one for 2 5
and 10 and then we'll expand on them to
so here's a list of divisibility rules
and we've already talked about them for
the numbers 2 5 and 10. and it turns out
the rules are relatively simple a number
is divisible by 2 if it ends in a 0 2 4
whoops that's supposed to be 6 or 8.
so for example you could say oh well 2
divides into four thousand seventy six
it ends in A6 and therefore 2 will
evenly divide into it
a similar rule to this is does 5 go into
a number and the rule is simple if the
number ends in a zero or a five then
five must divide that number
so that is you could say 5 divides 99 000
000
55. like that would be a true statement
and another one that is similar to that
again was for the dumper 10.
10 divides into a number if that number
ends in a zero so for example I know 10
will divide 111
330 like this would be true
now there's also rules for three four
six eight nine and twelve and let's
discuss those now I'll try to group them
into similar types of rules
let's look at the number four
how can you tell if four evenly divides
into a number
the rule for this will be similar to
that of 2 5 and 10 but we'll have to use
the last two digits of the number
instead of just the last digit itself
so the divisibility rule for 4 is 4
divides into a number if the last two
so an example for this would be
does 4 divide into three thousand twelve
is this a true statement and the answer
is yes and the reason for this is if you
look at the last two digits 12 well 4
divides 12. in fact it goes in three times
times
so this saves you a little bit of
trouble instead of dividing this whole
thing out you can say well does 4
actually divide in just the last two numbers
numbers
a rule that is similar to 4 is that of 8.
8.
this time we're not going to check the
last two digits but we'll look at the
last three digits
that is a number is divisible by eight
if the last three
three digits
digits are
are divisible
by 8.
so again this can save you checking the
entire number and just focus on the last
three digits
for example does 8 divide into nine
thousand eight hundred eight is this
true and the answer is going to be yes
if you look at the last three digits
which is 808
definitely goes into that 101 times so
just by looking at the last three digits
I know that 8 must go into it
so to recap for two four five and ten we
could figure out just by looking at the
last digit whether these numbers went
into it
for four and eight those are somewhat
similar for four you have to look at the
last two digits and for eight you have
to look at the last three digits
I think the next two that are similar
are that of three and nine so let's look
at the divisibility rule four three how
do you know if three goes into the number
number
the rule for this is three goes into a
number or a number is divisible by three
of the digits
by three
so here's an example for this one
does three divide into the number 795.
795.
the way you can check this is by taking
the numbers seven
nine and five and adding them together
and they sum to 21.
well 3 evenly divides into 21. and
because that is true according to this
rule we know that 3 must also divide
into 795 so this is a true statement
again the reason for that is if you sum
these digits and three divides it which
it does then 3 must divide this number here
here
the rule for nine is similar it is if
the sum of the digits is divisible by
nine so let's write that one down next
so a number is divisible by 9 if the sum
of the digits
digits
by 9.
a quick example for this one is does 9
divide into the number
189 is this true well if you look at one
plus eight plus nine this sum is 18 and
9 does in fact go into 18 twice
therefore 9 must go into 189 as well
now this leaves us with two other rules
uh one four if a number is divisible by
6 or 12.
now the way this one's going to work is
if the number is both divisible by two
and three
then it must be divisible by 6 as well
so this rule actually depends on two
other rules the two numbers that in fact
multiply to six so for this rule here A
number is divisible by three if it is
and this leaves us with the final rule
that is for when is a number divisible
by 12. this is going to be a similar
rule to that of 6. if a number is
divisible by both 3 and 4
then it is also divisible by 12.
so I'll state that here
is divisible
by both
three and four the two numbers that
actually multiply to 12.
okay so these are all the rules let's
in example two I have three numbers
listed here I want to use the rules on
the previous slide to determine whether
these numbers divide into these numbers
and let's start with 360.
uh by the way 360 is the number of
degrees in one rotation if you've ever
wondered why that is true maybe we'll
get some insight into that once we check
if these numbers divide into this one
so let's uh just go through the list
does two divide
divide
360. and the answer is yes and that is
because it ends in zero
so let's put after this yes it ends
in zero which is an even number
there's nothing else to check let's move
on to three does
does
three divide into Three Sixty is this true
true
well to determine if this is true the
rule for three said
you add the digits which is three plus
six plus zero and this sums to 9.
well 3 divides nine
so if 3 divides the sum of the digits it
must divide the number itself
okay moving on to four
does four divide 360
. now the rule for four was if the last
two digits are divisible by four then
the entire number is divisible by four
and we can actually work this one out
does 4 divide into 60 I believe it goes
in 15 times so here it'll go in once
bring down the two it goes in another
five times with a remainder of zero so
yeah four divides sixty and for this
reason it divides the entire number 360.
okay uh moving on to five
does five divide 360 and the answer to
this is yes
um 360 ends in a zero and the rule for
five was if nsyn is zero or a five then
five must divide it
all right so for six does six divide 360.
360.
um oh the rule for this one was if both
2 and 3 divided then six divides it as
well well we just showed earlier that
well I know two divides 360. and three
divides 360. and since both of those are
true it must be that 6 divides 360 as well
well
all right moving on to eight we don't
have a rule for seven
the rule for eight was if the last three
digits divide 360 then eight divides
360. well unfortunately there are only
three digits in this number so I think
we'll have to go through the process of
actually dividing this one out so does
eight go into 360 it doesn't go into
three but it goes into 36 four times
and then there's four left over bring
down the zero and eight times five is forty
forty
so yeah eight goes into the last three
digits which is actually the number
itself so okay eight divides Three Sixty
the rule really didn't help us out too
much in this problem
um what about nine
does nine divide 360
. I think this is going to be true and
the rule for nine was if nine divides
the sum of the digits
then it divides the entire number itself
so if I look at that down here I know
three plus six plus zero is Nine and
Nine definitely divides nine so we'll go there
there
uh we have two more I'll try to fit that
in here does ten divide 360 and I know
the answer to this is yes because the
number ends in a zero and there is one more
more
twelve divide 360 and the answer to this
one is yes as well and the reason for 12
was if both 3 and 4 divided then 12 must
divide it two so I'll just note
afterwards here that well 3 divides 12
and 4 divides 12. if both of those are
true well then it divides 12 divides
into it as well
so the reason why 360 is a good choice
for the number of degrees in a circle is
a lot of numbers evenly divide into it
and therefore it makes arithmetic with
this number very easy the concept of 360
degrees in a circle was developed a very
very long time ago long before the time
of calculators and it's a nice number to
work with because you can divide it
evenly with a lot of numbers
okay so let's do this again but for um
um
2022 which we can consider this like hey
the year 2022.
um does 2 divide into this number
so does 2 divide 2022 and the answer is
yes and the reason for that is well it
ends in an even number a two
all right does three divide into 20 22
and the way that we checked this was by
adding the digits which is really just 2
plus 2 plus 2. I don't really need to
add the zero in and we get six
3 divides six therefore 3 must also
divide the number 20 22.
okay what about four does four divide 20 22.
22.
I think this is going to be no
in fact if you want you could put a line
through this and say 4 does not divide
into 20 22. the reason is if you look at
the last two digits 22
4 does not divide into 22. it divides
into 20 it divides into 24 but it does
not divide into 22. so okay we've only
ranked one where it does not work
but actually
um moving on to 5 the number doesn't end
in a zero or a five so it's enough to
say well 5 doesn't divide into 20 22
either and the reason for that is if you
look at the last number it is a 2 it
would have to be a zero or a five
if we look at six well both 2 and 3
divide into this number and therefore we
can say well 6 must divide 20 22. and
the reason for that is both two
and 3 divided
so those are both true
if we look at
eight does eight divide into 2022 to
figure this out we have to see if 8
divides into the last three digits which
is really just 22.
does eight
divide into 22
that's not going to work I know that it
goes into 16 and it goes into 24. if you
try to divide 8 into 22 it goes from
three times you get 16 but then you have
a remainder of 6 left over and it should
have been a remainder of 0 but it wasn't
so therefore the answer here is no it
did not work so 8 does not divide into 2022.
2022.
okay what about if we look at 9 does
that divide into 2022
well we already know the sum of the
digits is 6. does 9 divide into six no
so again here if you add up the non-zero
digits you get six but nine is bigger
than 6 and therefore in this case does
not divide into it so again here we have
a nope
no good you could say 9 does not divide
into 20 22. so
moving on to 10
does that divide into 2022 also no the
last digit is a 2 and it was supposed to
be a zero so that's not true
and the last one is 12.
does 12 divide into 2022 and the answer
here is also know that this does not
work and the issue is 3 divides into it
but 4 does not both of these numbers
have to divide into 12 for 12 to work
but since 4 does not neither does 12 so
this did not work so here we actually
ran into some where there was a problem
it did in fact not work
all right so we have one left here it's
this massive number what is this even
this is million billion trillion
quadrillion nine quintillion I mean this
is a massive number if you were to
divide these out by hand it would take a
very long time but there's a lot of
zeros in here so I'm thinking well these
rules will make figuring out if they
divide into here much faster so let's
start with uh two
also to save some time I'm just going to
put two with a question mark after it so
we're not writing this massive number
over and over again does 2 divide into
it and the answer is yes and the reason
is it ends in an 8.
ends in eight
does 3 divide into it well if we sum the digits
digits
that is let's see I'll put it below here
that's nine plus two there's a bunch of
zeros and then we have eight plus eight
so that is 16 17 18
and 9 is 27 3 does divide into 27 so
therefore the answer to this one is yes
that is because 3 divides into the sum
of the digits and some of the digits
happens to be 27.
okay does 4 divide with this number for
this we look at the last two digits of
the number which in this case is just
the number eight itself
um does 4 divide into eight and the
answer is yeah and since 4 divides into
the last two digits four divides into
the entire number
uh does five work no no good it ends in
an eight not a zero or a five
does it work for six
and the answer is yes
and that's because both two and three
worked and therefore 6 must work as well
does eight divide into it for this you
have to look at the last three digits so
we're basically asking the question does
eight divide into 808 and we ran this in
the last slide it certainly does it goes
in 101 times and therefore the answer to
uh does it work for 10.
and the answer is no again it ends
in eight instead of a zero
I actually skipped over nine let's
consider that next
well we already have the sum of the
digits the sum of the digits is 27 and
it turns out that 9 does divide into 27
it goes in three times and because that
happens I know the answer to this is yes
there's one left over and that is 12 and
the answer here is also yes and that's
because both 3 and 4 divided into the
number and therefore so must twelve
so this took a while to get through all
these numbers but this was much faster
than using long division or even trying
in the second part of this lesson we'll
consider prime numbers so let's start
with its definition a prime number is
any natural number a positive integer
whose only divisors are one and itself
and for theoretical reasons we exclude
the number one
so an example of a prime number is five
five is prime
and the reason for this is one divides
into five five divides into five but
nothing else divides into five
an example of a number that is not prime
is let's say six
and this is because well one divides
into six and six divides into six but
the issue is both two and also three
divide into six and the second one of
those happens we know six is not Prime
in fact you can factor six you can say
six is equal to two times three
we call this a prime factorization of
this number here that is you split the
number apart using prime numbers both 2
and 3 are Prime and it's practice equal
to 6.
what's the prime factorization of five
it's actually just 5 itself because it
doesn't Factor at all
another number that is not prime is uh
and the reason for this is well 1
divides into 9 no problem nine divides
into 9. but 3 divides into 9.
in fact the prime factorization for nine
is three times three
so here you can actually break apart
nine into a product of just prime numbers
numbers
and you can also Express 3 times 3 using
exponents which is three squared so here
we call 3 squared the prime
factorization of nine a very important
fact in number theory is that if you
take any natural number and find its
prime factorization it is unique that is
if you find the prime factorization of
nine there's only one choice it'll only
ever be three squared it won't ever be
anything else same thing for six it's
two times three there's no other way to
factor it using primes and five well
five is just five and so this is its
prime factorization
in example three I've listed out some
numbers that are larger than the ones
that I put up here we'll like to do is
try to find the prime factorizations of
these numbers and we'll express any
repeated primes using exponents and by
that if you have two primes in a row
instead of doing three times three use
an exponent to say you have two of those numbers
numbers
let's start with
147. I'll go over a systematic way that
you can find the prime factorization of
this number
we first need to know a number that goes
into 147
I know 2 5 and 10 don't go into it but I
have to find something
if you look at the sum of the digits
it's one plus four plus seven that's 12.
if the sum of the digits is 12 well 3
goes into 12 therefore 3 must actually
divide this number
so to find a prime factorization you
have to use division to split this thing
apart so I know for a fact and I'll put
this down here I know 3 has to evenly
divide into 147 and the reason for this
is if I sum these digits I get 12 and 3
goes into 12.
just doing the math on this
okay 3 goes into 14
4 times because you get 12.
you get 27 and so 3 goes into 27 9 times
perfectly and we have a remainder of 0.
this means we can go back up to 147 and say
say
okay I can break this apart as 3 and 49. now
now
3 cannot be split apart anymore
but 4910 and I happen to know that 7
times 7 is 49.
3 and 7 are prime numbers
it should be that the product of these
values 7 times 7 times 3 is equal to
147. therefore this is the prime
factorization you could say it's 3 times
7 times 7.
and we have two sevens in a row let's
use an exponent of two to more concisely
write the prime factorization as follows
it's 3 times 7 squared
so here's a nice systematic way to
determine the prime factorization of a
given number
try to find something that divides into
it and we have a bunch of divisibility
rules that tells us like a place to start
start
divide into it and then try to break
apart that smaller number
then multiply all the final numbers
together to get your final answer
all right let's try that again for 120.
so we have to think of a number that
goes into it well it ends in a zero so 2
goes into it but so does Five and
actually so does 10.
if we split this apart I think I'll use
10 because it's the largest value this
is 10 times 12.
I'm now going to split apart both 10 and
12 because neither of those are prime
let's start with uh that was supposed to
be 10.
10 is 2 times 5 and both of those
numbers are prime
12 is 3 times 4
so I'm really just focusing on these
numbers Here If I multiply these numbers
together I'll get 120 2 is prime so is 5
and 3. but 4 is not prime we can break
that one apart into two times two
therefore If I multiply 2 times 5 times
3 times 2 times 2 that product is 120.
so we can say
120 is equal to let's see I have 2 times
2 times 2 so I'll put 2 times 2 times 2.
so this is the prime factorization these
are only prime numbers and here I have
three twos in a row I can write this as
2 to the power of 3 and this is my final
answer it's 2 to the power of 3 because
I have three twos in a row I then have a
3 and a 5.
for 330 again I need a number that
divides evenly into it and oh I see 10
will work again
so I know that
330 is 10 times 33.
and I know 10 can be split apart into 2
times 5.
now what about 33 well the sum of the
digits is six so I know 3 goes into it
in fact this is 3 times 11. if you
multiply a one and a one by a 3 you get
back to a 33.
so 2 is prime so it's 5 and 3 is 11 Prime
Prime
yeah nothing else will divide into this number
number
notice that all these primes are
different and their product is 330.
therefore the best we can do is say 330
is 2 times 3 times 5 times 11.
I like to write them in increasing order
all right last one 73.
okay the trick is to think of a number
that goes into it
so well I know two won't work
um five and ten don't work
and if two doesn't work 6 isn't going to work
work um
um
let's see the sum of the digits is 10
and 3 doesn't go into 10 so 3 doesn't
work okay um
um
4 does not divide evenly into 73 either
so I know 4 is not going to work
so I've checked two three four five six
eight doesn't go into it that's no good
9 doesn't work because the sum of the
digits is 10 and 9 does not go into 10. uh
uh
and yeah 12 doesn't work either because
both 3 and 4 actually don't go into it
so like none of these work
since none of these values work we can
keep trying but it turns out that there
is no other number besides 1 and 73 that
goes into 73. so what's the prime
factorization of 73 it actually is just
73 and the reason for this is because 73
so here we go
um for these four numbers these are
their prime factorizations and they are
unique there is no other way that you
can express the prime factorizations of
these numbers
one final thought
this is the reason why one is not
considered a prime number it's for this
idea of uniqueness like for example
um just to put this down here we said
147 was equal to 3 times 7 squared and
that's the only way you can write it
using prime numbers
if you allowed one to be a prime number
you could say well 147 is equal to 1
times 3 times 7 squared or you could say
it's 1 squared times three times seven
squared there's a lot of ways you could
write this
if you include one as a prime number
you no longer get this idea of a unique
way to express your prime factorization
if one is prime
what we do is we just say let's just not
use one as a prime number that'll mean
if you look at a prime factorization of
any natural number that is two or more
all right let's look at the definition
for the greatest common divisor
so the greatest common divisor of two
natural numbers is the largest natural
number that divides both original numbers
numbers
now that's quite a definition but let's
clear that by looking at example four I
want to find the greatest common divisor
of six and eight
the way to do this is you can take both
six and eight and you can list out all
the numbers that divide them
so the natural numbers that divide six
are one two three and six itself and
that's it
and the numbers that divide eight are one
one
two four and eight
so here we've listed out all the
divisors for each of these numbers individually
individually
when we say the greatest common divisor
if you look at both of these lists which
number is common to both lists and is
also the largest
well one is on both of the lists that's
great but 2 is on both of the lists as
well to divide 6 and 2 divides eight
that is the largest number for which
this is true so
so
2 divides eight two divide six it's the
largest number common to both so I'll
say 2 is the I'll say greatest common
divisor of six and eight nice
so if the numbers are relatively small
it's not too hard to figure it out you
can just list out to the divisors and
figure out which is the largest that is
common on both lists
well if the numbers are bigger like
let's say 84 and 56 what is the greatest
common divisor of these two
you could use the same procedure you can
list out all the divisors and then pick
the biggest one but there might be a
better way to go about doing this
so let's try this problem again but
we'll use a new approach
what I'm going to do is for the number
six I'm going to find its prime
factorization which I know six is two
times three
and therefore that's its prime factorization
factorization
it's 2 times 3.
now for eight
C 8 is
2 times 4.
and 4 is 2 times 2
that means the prime factorization of
eight is two times two times two that's
two cubed you have three twos in a row
you can actually look at these prime
factorizations to find the greatest
common divisor
what we will do is gather up the prime
numbers that are common to both of these factorizations
factorizations
so over here 2 divides six and so does 3
. notice over here 3 does not divide
into eight
this means if you want something common
that divides both eight and six you can
only use two you can't use three so in
constructing your greatest common
divisor you have to use 2 as a prime
number you can't use three because three
does not divide into eight so you're
only allowed to use two
um notice there's no other prime numbers
to consider there's just twos and threes
the next question is well how many twos
are you allowed to use
one two can divide into six but three
twos can divide into eight and you want
something that is going to work for both
well since you can only divide two into
six once you can't use the second or a
third time
this means two just by itself here is
the greatest common divisor so again
just to reiterate you go through each
Prime on each list you're only allowed
to use the primes that are common on
both lists
and the next thing you have to do is use
the smallest amount of primes between
the two so here I can only use two to
the first because if I use 2 to the
second or third that won't divide into
six and that is how you find the
let's try that with these two numbers
here 84 and 56.
what I'll do is start by trying to
figure out what is the prime
factorization of 84.
so let's see
I know that 4 goes into this number and
and
that is 4 times 21.
now I can split 4 apart as 2 times 2 and
these are both primes so that's where
this process ends
but for 21 I can write that as 3 times 7.
7.
now 3 and 7 are both Prime and therefore
we have our prime factorization it's 2
times 2 times 3 times 7. that means we
can write 84 as 2 squared because
there's two of them times three times seven
seven
okay I'll do this again but for 56.
so I'll put that down here for 56. okay
okay
um I know 56 is 8 times 7.
7 is prime so I can break apart eight
and not seven
8 is 2 times 4 and as we said 4 is 2
times 2.
so 56 is 2 times 2 times 2 times 7. that
is this is equal to 2 cubed because we
have three of them times seven
it's now time to take this information
and construct the greatest common divisor
divisor
all right what we do is we look at the
lists and we're only allowed to use
primes that are common to both 3 divides
into 84. 3 does not divide in 56 so
we're not allowed to use three notice
that 2 and 7 are common to both lists so
we are allowed to use a 2 and we are
allowed to use a seven
here 7 appears once and here it appears
once so it's actually the same number
for both so I'll just have 7 to the first
first
but what's the largest number of twos
that you're allowed to use here there's
two of them and here there's three
well if you want this number to divide
both of these you can't use this amount
here it's too much 2 cubed won't go into
84. but 2 squared well in fact it'll go
into this number two therefore if you
take the smaller of the two exponents
it's 2 squared
I think we should multiply this out 2
squared is four
times seven is twenty-eight
28 is the greatest common divisor of 84
meaning it goes into 84. it goes into 56
and there's nothing bigger than this
a related definition to greatest common
divisor is that of a least common multiple
multiple
by definition the least common multiple
of two natural numbers is the smallest
number smallest this time that is
divisible by both of the original numbers
numbers
now to make sense of this definition in
example four let's find the least common
multiple of 5 and 10.
so this time we're talking about
multiples instead of divisors what does
that mean
if you start with 5 and you list out the
multiples that means you're just
counting by the number five so five 10
15 20 25 and so on
these are all multiples of five because
this is five times one five times two
five times three and so on
we can do the same thing for 10.
so the multiples of 10 are 10. 20. 30 40
so here we have two lists multiples of
five and ten
and notice that there are numbers that
are common on both lists like here
there's 10 and here there's 10. so I'm
going to highlight both of those this
means they're common multiples
um also there's well
20 and 20 they'll be 30 and 30.
the smallest of them is 10. so there are
multiples that are common on both lists
but the smallest the least common
multiple which I'll abbreviate l c m in
this case is equal to 10.
so that's the basic definition for least
common multiple it's not too bad if the
numbers are small but what if the
numbers are larger like 144 and 300
I see it says greatest common divisor
here but really I'd like to find the least
least
common multiple of 144 and 300.
before trying this one let's go back to
5 and 10 and take a second approach for
finding the least common multiple and
we'll do this again by using the prime
factorizations of these numbers
so to start with five what is the prime
factorization of five
well you can't really do much else
besides call it five times one and so it
is its own prime factorization is five
and for 10
well that is 2 times 5 and well that's
the best you can do
if you're looking to construct a least
common multiple from both of these prime
factorizations the way you'd go about it
is by saying well it has to be a
multiple of both numbers
therefore we have to use every single
prime number that appears last time we
only took the ones that were common to
both here we have to use every single
prime number
so let's see we have to use both 2 and
5. those are the only two prime numbers
that appear
last time we were restricted and we had
to take the smallest number of the
exponent that appeared or this time
we're going to choose the biggest since
we want it to be a multiple of both of
the numbers so here I'll use 2 to the
first even though there are no twos over
here there's a five to the First on both
of these they're the same but that means
we have to use five and what do you know
two times five is equal to 10.
let's try this process on 144 and 300.
to find the least common multiple we'll
find their prime factorizations and then
we'll carefully choose the appropriate
number of prime numbers to construct the
least common multiple
let's start with um 144.
144.
if we're looking for the prime
factorization the last two digits are
divisible by four so I know 4 divides
into this number
that is equal to 4 times 36.
now 4 is 2 times 2 and those are both
Prime and 36 is 6 times 6.
and 6 you can break apart into two times
three and two times three
so this means the prime factorization of
144 is 2 times 2 times 2 times 2 there's
four twos
that is 2 to the fourth power
and then we have 3 times 3 so 3 squared
and that is it for the prime
factorization of 144.
now what about the prime factorization
of 300.
a lot of ways to break this apart
um well this is three times a hundred
I can't break apart three but I can
break apart a hundred into four times
twenty five
four is two times two and twenty-five is
five times five
okay these are all prime numbers we have
two times two
that is
2 squared
there's a single three and we have 5
times 5 which is five squared
we can now take this information to
construct the least common multiple
if you want this to be a multiple of
both of these numbers we have to use
every single Prime that appears on these
lists so like here if you want to be a
multiple of 144 you gotta have twos and threes
threes
and if you want to be a multiple of 300
you have to have twos and threes we
actually also have to have fives as well
so I know we have to use
twos threes and fives
we just want to choose the appropriate
number of exponents for all these
numbers so that we're not just a common
multiple but we're a least common multiple
let's see so here's two to the fourth
and here's 2 squared
you have to use two to the fourth if you
ever want to multiply to 144 therefore
between four and two you want to choose
the larger exponent so I'll put a power
of 4 over here
so this is 2 to the fourth
next we have 3 squared and we have 3 to
the first
you have to use 3 squared if you want to
multiply to 144. if you just have 3 to
the first you can multiply to 300 but
the smallest number that's going to work
for both is 3 to the power of 2. so
basically I looked at the exponent here
which is 2 there's an exponent of 1 here
and I chose the larger of the two
there are no fives here but there's a 5
squared down here
if you want to be a multiple of 300 you
have to have 5 squared and therefore
this is 5 squared over here
if I simplify this number
2 to the fourth is 16.
16.
3 squared is 9 and 5 squared is 25.
I think I'm going to multiply this out
on a calculator 16 times 9 times 25 is
so what does this mean if you list out
the multiples of this number and the
multiples of this number on a list
this is going to be the first number on
that list that is a multiple of both
in example five we'll look at an
application of least common multiples
so the question says a movie theater
runs films Non-Stop
there's an action movie that runs for
115 minutes and a horror film that runs
for 70 minutes and we'll assume that
there's a 10 minute break at the end of
each film until it starts again
well if that happens if they both start
at 11AM when will the two movies start
again at the same time
to see why this is going to be least
common multiple type of question I'm
going to focus on the action movie first
so Louis starts it runs for 115 minutes
and then there's another 10 minutes that
you have to add on to that until the
movie starts again
so it's really 125 minutes
so the movie plays 125 minutes later it
plays at the start again
if you play it one more time there's
going to be 250 minutes twice as long
until the movie plays again
and then if it plays three times then
it's going to be 375 minutes until it
plays again and so on
we can do the same thing for the horror
movie it's a 70 minute film
which is pretty short and then there's a
10 minute break so that's a total of 80
minutes so the movie starts and then 80
minutes later it starts again
so it starts eight minutes later it
starts again then 160 minutes starts
again and then 240 minutes later it
starts again and so on
if we want to know when these two movies
are going to start at the same time
again we're looking for a number on both
of these lists that appears for the
first time and that is exactly what a
least common multiple is
now these numbers are pretty big I think
a faster approach will be to take a
number like 125
find the prime factorization of it and
then use that to find the least common multiple
multiple
so for 125
that is
25 times 5. I'll put 5 times 25.
and then
25 is 5 times 5.
so the prime factorization of this
number is 5 cubed
I'll do the same thing for 80 minutes
so we have an 80 minute film
but if I find the prime factorization
this is 10 times 8
10 is 2 times 5.
8 is 4 times 2 and 4 is 2 times 2.
so this is going to be 2 times 2 times 2
times 2 that's 2 to the fourth
and we also have 5 to the first Power
and that's it so here's the prime
factorization of this number
we are looking for the least common multiple
multiple
this means we have to use every single
Prime that appears on both lists which
is a 2 and a 5. the reason for this is
if you don't include a two you'll never
be a multiple of this number so you
can't just use fives we have to use both
five and two
now I want to use the biggest exponent
that appears on either of the lists
two to the fourth is the largest so I
have to put a 4 here
that's the only way we'll be a multiple
of this number
and I also have to use 5 cubed
if I want to be a multiple of 125.
so whatever this number is is the least
common multiple of this amount of time
and this amount of time
and 2 to the fourth is 16 5 cubed is
125. I'm going to do that in my
calculator is 16 times 125 is 2 000.
000.
so these films are playing Non-Stop
and this is 2 000 minutes
and that's going to be the first time
that these two films start at the same
time again so basically
11 am is here
and then okay they don't match up they
don't match up they don't match up but
after 2 000 minutes they do match up again
again
so I'll do is figure out well how many
hours is this and then I'll count after
11 A.M so okay
2 000 minutes I have to convert that
into hours
so I'll take 2
000 minutes and divide that by 60
because there's
60 minutes in an hour and that is
33 and one third hours
now what's a third of an hour that is 20 minutes
minutes
so this is 33 hours and 20 minutes
20 minutes is a third of an hour because
20 plus 20 plus 20 is 60 minutes
so I guess one way you can answer this
problem is by saying well after this
amount of time the films will start at
the same time again
but if you'd like to actually have a
time like this I think what I'll do is
I'll start with
red 11 am
am
and if we add 24 hours to this we'll be
back at 11 A.M
and we've taken 24 hours off of this 33
hours so if we have 33 hours
and 20 minutes and we subtract 24 hours
that means there's still another
and 20 minutes to go
this will mean you've got a total of 33 hours and 20 minutes okay so
hours and 20 minutes okay so if you add one more hour that takes you
if you add one more hour that takes you up to noon
up to noon so now we're down to eight hours so
so now we're down to eight hours so maybe I'll say okay if we go up to 12
maybe I'll say okay if we go up to 12 p.m that is now 8 hours and 20 minutes
p.m that is now 8 hours and 20 minutes remaining
remaining and that means the next showing will be
and that means the next showing will be at
at eight
eight twenty
twenty P.M
P.M that is very confusing to get to this
that is very confusing to get to this time but essentially I figured out the
time but essentially I figured out the total number of hours until we start
total number of hours until we start again I went to 11AM
again I went to 11AM I went 24 hours so we're at 11 A.M again
I went 24 hours so we're at 11 A.M again and then there was nine hours and 20
and then there was nine hours and 20 minutes remaining I took another hour to
minutes remaining I took another hour to get us up to 12 p.m and then eight more
get us up to 12 p.m and then eight more hours to get us up to 8 p.m and then
hours to get us up to 8 p.m and then there's still this 20 minutes left over
there's still this 20 minutes left over so this would be the actual time
so this would be the actual time of course this assumes that the theater
of course this assumes that the theater is open 24 hours a day but in any case
is open 24 hours a day but in any case uh this is the value that we're looking
uh this is the value that we're looking for
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