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CCSS.Math:

William and Luis are in
different physics classes at Santa Rita. Luis's teacher always gives
exams with 30 questions on them, while William's
teacher gives more frequent exams with
only 24 questions. Luis's teacher also assigns
three projects per year. Even though the two classes
have to take a different number of exams, their
teachers have told them that both classes-- let me
underline-- both classes will get the same total number
of exam questions each year. What is the minimum
number of exam questions William's or Luis's class can
expect to get in a given year? So let's think about
what's happening. So if we think about
Luis's teacher who gives 30 questions per test,
so after the first test, he would have done 30 questions. So this is 0 right over here. Then after the second test,
he would have done 60. Then after the third test,
he would have done 90. And after the fourth test,
he would have done 120. And after the fifth test,
if there is a fifth test, he would do-- so this is if
they have that many tests-- he would get to 150
total questions. And we could keep
going on and on looking at all the multiples of 30. So this is probably a hint
of what we're thinking about. We're looking at
multiples of the numbers. We want the minimum multiples
or the least multiple. So that's with Luis. Well what's going
on with William? Will William's teacher,
after the first test, they're going to
get to 24 questions. Then they're going to get
to 48 after the second test. Then they're going to get
to 72 after the third test. Then they're going to get to 96. I'm just taking multiples of 24. They're going to get to
96 after the fourth test. And then after the fifth test,
they're going to get to 120. And if there's a sixth test,
then they would get to 144. And we could keep going
on and on in there. But let's see what
they're asking us. What is the minimum
number of exam questions William's or Luis's class
can expect to get in a year? Well the minimum
number is the point at which they've gotten the
same number of exam questions, despite the fact
that the tests had a different number of items. And you see the point at which
they have the same number is at 120. This happens at 120. They both could have
exactly 120 questions even though Luis's teacher
is giving 30 at a time and even though William's
teacher is giving 24 at a time. And so the answer is 120. And notice, they had a
different number of exams. Luis had one, two,
three, four exams while William would have to
have one, two, three, four, five exams. But that gets them both
to 120 total questions. Now thinking of it in terms
of some of the math notation or the least common multiple
notation we've seen before, this is really asking us what is
the least common multiple of 30 and 24. And that least common
multiple is equal to 120. Now there's other
ways that you can find the least common multiple
other than just looking at the multiples like this. You could look at it
through prime factorization. 30 is 2 times 15,
which is 3 times 5. So we could say that 30 is
equal to 2 times 3 times 5. And 24-- that's a different
color than that blue-- 24 is equal to 2 times 12. 12 is equal to 2 times 6. 6 is equal to 2 times 3. So 24 is equal to 2
times 2 times 2 times 3. So another way to come up with
the least common multiple, if we didn't even do this
exercise up here, says, look, the number has to be
divisible by both 30 and 24. If it's going to
be divisible by 30, it's going to have to
have 2 times 3 times 5 in its prime factorization. That is essentially 30. So this makes it
divisible by 30. And say, well in order
to be divisible by 24, its prime factorization is
going to need 3 twos and a 3. Well we already have 1 three. And we already have 1 two,
so we just need 2 more twos. So 2 times 2. So this makes it--
let me scroll up a little bit-- this right over
here makes it divisible by 24. And so this is essentially
the prime factorization of the least common
multiple of 30 and 24. You take any one of
these numbers away, you are no longer going to be
divisible by one of these two numbers. If you take a two away, you're
not going to be divisible by 24 anymore. If you take a two
or a three away. If you take a three
or a five away, you're not going to be
divisible by 30 anymore. And so if you were to
multiply all these out, this is 2 times 2 times 2 is 8
times 3 is 24 times 5 is 120. Now let's do one more of these. Umama just bought one
package of 21 binders. Let me write that number down. 21 binders. She also bought a
package of 30 pencils. She wants to use all of
the binders and pencils to create identical
sets of office supplies for her classmates. What is the greatest
number of identical sets Umama can make using
all the supplies? So the fact that we're
talking about greatest is clue that it's probably
going to be dealing with greatest common divisors. And it's also dealing with
dividing these things. We want to divide these
both into the greatest number of identical sets. So there's a couple of ways
we could think about it. Let's think about what the
greatest common divisor of both these numbers are. Or I could even say the
greatest common factor. The greatest common
divisor of 21 and 30. So what's the largest number
that divides into both of them? So we could go with
the prime factor. We could list all of
their normal factors and see what is the
greatest common one. Or we could look at the
prime factorization. So let's just do the prime
factorization method. So 21 is the same
thing as 3 times 7. These are both prime numbers. 30 is, let's see,
it's 3-- actually, I could write it this
way-- it is 2 times 15. We already did it
actually just now. And 15 is 3 times 5. So what's the largest
number of prime numbers that are common to both
factorizations? Well you only have a
three right over here. Then you don't have a
three times anything else. So this is just going
to be equal to 3. So this is essentially
telling us, look, we can divide both
of these numbers into 3 and that will give
us the largest number of identical sets. So just to be clear
of what we're doing. So we've answered
the question is 3, but just to visualize
it for this question, let's actually draw 21 binders. So let's say the 21 binders so
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21. And then 30 pencils, so
I'll just do those in green. So 1, 2, 3, 4, 5,
6, 7, 8, 9, 10. Let me just copy and paste that. This is getting tedious. So copy and paste. So that's 20 and then
paste that is 30. Now, we figured out that 3
is the largest number that divides into both
of these evenly. So I can divide both of
these into groups of 3. So for the binders, I could
do it into three groups of 7. And then for the
pencils, I could do it into three groups of 10. So if there are
three people that are coming into
this classroom, I could give them each seven
binders and 10 pencils. But that's the greatest
number of identical sets Umama can make. I would have three sets. Each set would have seven
binders and 10 pencils. And we essentially
are just thinking about what's the number that we
can divide both of these sets into evenly, the
largest number that we can divide both of
these sets into evenly.