MTEL Math Practice Test: 12 -15

Problem 12.

The prime factorization
of a natural number.

You can kind of view that
as positive integers.

Sometimes they'll throw a 0 in
the natural number definition,

depending on what context
you're working in.

But the prime factorization of
a natural number n can be

written as n is equal to pr
squared, where p and r are

distinct prime numbers.

Fair enough.

How many factors does n have,
including 1 and itself?

Well, they're saying that this
is some natural number that

can be written this way.

Let's just pick one example.

So if we just set p-- I'm just
going to pick some prime

numbers here-- p is equal to 2
and r is equal to 3, then n is

equal to p times r squared,
times 3 squared, which is

equal to 2 times 9, which
is equal to 18.

This is just one instance--
one n that satisfies these

conditions.

I could have picked 5 and 7, or
I could have picked 3 and

2, but all of them should have
the same answer, because

that's what this question
is implying.

So this is a possible n.

And then how many factors
does it have?

Let's factor it out.

It's got 1 and 18,
2 and 9, 3 and 6.

So it's got 1, 2, 3,
4, 5, 6 factors.

So the answer is D.

And you could try it out with
other numbers, if you like.

Make p is equal to-- make
p equal to 3, and

make r equal to 2.

Then you'd have 3 times
2, times 2.

You'd have the number 12, and
you'd see its factors are 1

and 12, 2 and 6, and 3 and 4.

So you'll see the same thing,
exactly six factors.

Problem 13.

Given pn is equal to 150, where
p is a prime number and

n is a natural number, which of
the following must be true?

So, let's just think about it.

What are all the possible
values for p?

To do that, we'd have to
figure out the prime

factorization of 150.

I'll do that in black.

So 150, let's do its prime
factorization.

That's 2 times 75.

75 is 3 times 25.

25 is 5 times 5.

So 150 is equal to 2 times
3, times 5, times 5.

That's its prime
factorization.

So p could be any of
these numbers.

So for example, p could be-- if
p was 2, then n would be 3

times 5, times 5.

If p were 3, then n would
be 2 times 5, times 5.

If p were 5, then n would be
2 times 3, times 5, right?

Let me write this, make this
a little bit clearer.

These are my p's, because p
has to be one of the prime

factors of 150, and then n is
whatever's left over when you

divide by that p, right?

In each of these cases.

So let's look at the choices.

Which of the following
must be true?

p is a factor of either
10 or 15.

Let's see, 2 is a
factor of 10.

3 is a factor of 15.

5 is a factor of
both 10 or 15.

So that actually looks like
the correct answer.

Let's write that, let's
square that right now.

Let's look at the other choices,
just to make sure we

haven't missed anything.

10 is a factor of n.

Well, let's see, over here--
well, no, 10 is not a

factor of this n.

3 times 5, times 5 is 75.

10 is not a factor of this
number, so it's not going-- 10

is a factor of these two
choices, but this is a

possible n right here.

75 could be n.

So this is not the
correct answer.

n is a factor of either
10 or 15.

n is a factor.

Remember these are our
n's right here.

These are the potential n's.

This is not a factor of--
this is neither a

factor of 10 nor 15.

10 and 15 are smaller numbers
than all of these over here.

So this is not the answer.

And then 15 is a factor of n.

15 is a factor of n.

So in order to have 15 to
be a factor, you have

to have a 3 in there.

Now you look at this
choice, right here.

2 times 5, times 5.

This is 50.

This n right here is 50.

15 is not a factor of 50.

So our first inclination
was correct.

p is a factor of either
10 or 15.

Next problem.

All right.

The greatest common factor
of n and 540 is 36.

Which of the following
could be the prime

factorization of n?

So let's just think about
this a little bit.

The greatest common factor of
n and 540, so the largest

number that goes into
both 540 and this

mystery number n is 36.

Which of the following
could be the prime

factorization of n?

Let's just take the prime
factorization of everything,

just so we can kind of go down
to the most digestible parts.

So the prime factorization of
540, let's see, 2 times 70.

This is the same thing as 3
times 90, and 90 is 3 times

30, and 30 is 3 times
10, 10 is 2 times 5.

So the prime factorization of
540 is equal to-- there's two

2's here-- so it's equal to
2 squared times 3 to the

third, times 5.

That's 540's prime
factorization.

Now, the greatest common factor
of n and 540 is 36.

36 is equal to what?

That's-- I'll do it right
here-- 36 is 2 times 18.

That's 2 times 9, and
that is 3 times 3.

So 36 is equal to 2 squared
times 3 squared.

So, n has to be some multiple
of this that does not

still go into 540.

Let me show you what I mean.

Let's see which of these
could be possible prime

factorizations of n.

So A is 2 times 3 squared.

Well, that doesn't work because
remember, n, this

right here, 36 has to be
divisible into our n.

36, or 2 squared times 3 squared
is not divisible into

this, because we don't have
a 2 squared here.

So A is not our choice.

I'll do it in red when
I knock out choices.

So A is not our answer.

2 squared times 3 squared.

This is tempting, because 36
definitely goes into 2 squared

times 3 to the third.

Sorry, this is 2 squared
times 3 to the third.

2 squared times 3 squared
definitely goes into that.

In fact, it goes into
it three times.

This is 36 times 3 right
here, right?

Because the difference between
this number and this number is

the power of 3, and this one
has a higher power of 3.

So 36 definitely goes
into this number.

But we have to be
very careful.

This number, this number right
here, also goes into 540.

If n was this number right here,
then n would go into

540, because this is 2 squared
times 3 to the third, 2

squared times 3 to the third.

Then the greatest common factor
of n and 540 would be

this, instead of this.

So this can't be our answer
because if this was our n,

then n would be the greatest
common factor of n and 540.

So this can't be our choice,
because this is actually a

larger number that goes
into 540, than

this one, right here.

Let's look at choice C.

2 to the fourth times
3 squared, times 7.

This looks good, because
36 definitely

goes into this number.

2 squared definitely goes
into 2 to the fourth.

It goes into it four times.

3 squared definitely goes
into 3 squared.

And then we're multiplying
it by 7.

And what's good about this is
we're multiplying it by a

number or a factor that's
not in 540.

So this looks like our best
choice right here.

This looks like our
best choice.

Let's look at the last choice,
just to make sure we haven't

missed something.

2 to the fourth times
3 to the fifth.

This part, right here, is
definitely a multiple of our

greatest common factor, so
it looks good so far.

But then we do it times 5.

If this was our n, if this,
right here, was our n, then

the greatest common factor
of this number and 540

would not be 36.

It would be-- let's see, we
would have 2 to the squared,

because that's the largest power
of 2 that goes into both

of them, it would be 2 squared
times 3 to the third, times 3

the third right-- because that's
the largest power of 3

that goes into both-- times 5.

The greatest common factor of
this number and 540 is this,

which just happens to be 540.

So it can't be this
choice right here.

And you get to the same
argument on choice B.

The greatest common factor
of choice B and 540 is 2

squared-- because 2 squared goes
into both numbers-- times

3 to the third, which is a
very different number.

Or not a very different number,
it's 3 times 36.

So that's the other rationale
for why B is not

a legitimate answer.

Next problem, 15.

Scroll down a little bit.

A shipping container.

A shipping container measures 8
feet by 12 feet by 24 feet.

Let me draw that.

So we have 8 feet by 12
feet, and there's 24

feet, just like that.

So it maybe looks something
like that.

That's actually what shipping
containers really do look

like, looks something
like that.

The container is to be filled
with identical cube-shaped

boxes, each having
sides measuring a

whole number of feet.

Which of the following
expressions represents the

smallest number of such
identical boxes that could be

packed into the container with
no empty space remaining?

They say the smallest number
of identical boxes, which

means we want to get the biggest
possible, we need the

biggest possible boxes that
can be fit-- cubic boxes--

that can be fit into this
container with no space.

So the dimensions of those boxes
have to be the greatest

common factor of
8, 12, and 24.

So we could write this
right there.

Dimensions or, let me say the
box side, has to be equal to

the greatest common factor
of 8, 12, and 24.

So what's the biggest number the
goes into 8, 12, and 24?

You could do this
from inspection.

It's 4.

And you can verify that.

If the boxes were 4 on a side,
then in this dimension you'd

fit 8 divided by 4 of
them, or 2 of them.

So you'd have one, two boxes
in that dimension.

In this dimension, over
here, you would have

three of them, right?

12 divided by 4.

You'd have three boxes
in that dimension.

And in that dimension, you would
have-- let me draw these

just like that-- you
would have six.

1, 2, 3, 4, 5.

1, 2, 3, 4, 5, 6.

You would have 6, or
24 divided by 4.

So if you want to know the
total number of boxes you

fit-- I'm just looking at the
choices, they didn't actually

do the math-- they say, well,
in this dimension, I have 8

divided by 4 boxes.

In this dimension, I have
12 divided by 4 boxes.

And in this dimension, I have
24 divided by 4 boxes.

So that's how many
boxes we can fit.

We could multiply it out, but
this is actually one of our

choices, right there.

So the whole key is just to
realize, well, look, if the

boxes have to be cubes, and
there's no remaining spaces,

the dimension of the boxes has
to be divisible into all three

of these-- all three
of these-- numbers.

And I want the biggest possible
dimension because we

want the smallest number of
boxes, and that's where we got

the greatest common
factor of 4.

And then you say, OK, 4, and if
we have boxes of dimension

4 feet, and in this direction
we're only going to be able to

fit 8 divided by 4, this
direction 12 divided by 4, and

this direction 24
divided by 4.