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Current time:0:00Total duration:9:49

All right, we're on problem
125 on page 288. If r and s are positive
integers, is r/s an integer? Is r/s an integer? So that's really just another
way of saying, is s divisible into r? So let's see what the
statements are. Statement 1. Every factor of s is
also a factor of r. That answers our question. Every factor of s
is factor of r. Well, let me ask
you a question. What is the largest
factor of s? The largest factor of s is s. So this statement tells us that
since s is a factor of s, that s is also going to
be a factor of r. And something being a factor of
something means that it's divisible into it. So that means that s is
divisible into r. So this means that r/s
is an integer. So it answers our question. So statement 1, alone,
is sufficient. Statement 2 tells us, every
prime factor of s is also a prime factor of r. Every prime factor-- so let me
think of-- Immediately, I can think of a case where that
doesn't hold up. Where I could have every prime
factor being a factor of r. So let's say s is equal to 4. And its prime factorization
is 2 times 2. And let's say that
R is equal to 6. And its prime factorization
is 2 times 3. So we have a case here where
every factor, every prime factor of s, is a prime
factor of R. The only prime factor
of s is 2. And that's a prime
factor of R. But if we were to say R/s,
R/s would be 6/4, which is not an integer. So even though these satisfy
the second condition, this isn't an integer. But then I could have-- instead
of making R equal to 6, I could've made
R equal to 4. Sorry, I could have made R is
equal to 8, which is equal to 2 times 2 times 2. And in this case, it would
have been an integer. R/s would be equal to 8/4,
which is equal to 2. So statement 2, really doesn't
give us information as to whether r/s is an integer. So statement 1, alone,
is sufficient. Next problem. Switch colors. 126. If z to the n is equal to 1,
what is the value of z? z equals what? So statement 1 tells us,
n is a nonzero integer. n does not equal to 0. So let's think about this. For something to some power
to be equal to 1, what do we know about it? Well, if anything to the 0th
power is equal to 1, but they just told us that
it's nonzero. So we can't use this
condition. So that actually does restrict
z a good bit. So if n is a nonzero number,
what can I raise to the power to equal 1? Well, clearly 1, right? 1 to anything is going
to be equal to 1. But what other? Well, there's also the
possibility of negative numbers, right? Negative 1 to the nth power is
equal to 1, if n is even. And I think these are the only
two, if we take imaginary numbers out of the picture. And I think, on the GMAT, we
assume that we don't have any imaginary numbers. So assuming that, the only--
If we know that n does not equal 0 and z to the n is
equal to 1, the only possibilities that this allows
for is that z is equal to 1 or negative 1. And if it's negative 1,
then n would have to be an even number. But they didn't restrict
that yet. So statement 1, by itself,
it helps us. But it doesn't actually give
us enough information. We can just narrow it down to
z being 1 or negative 1. Let's see what statement
2 tells us. Statement 2 tells us,
z is greater than 0. So that, by itself, is
useless information. Because if z is greater than
0 and n can be 0, right? We're assuming we don't
have statement 1 yet. So if z is greater than 0 and
n could be 0, z could be a hundred to the 0th power. z could be a hundred
and that equals 1. z could be 99 to the
0th power, and that could be equal to 1. So z could be anything to the
0th power as long as it's greater than 0. So this, by itself,
doesn't help us. But if we use statement 2 and
statement 1 in conjunction, then it's interesting. Because statement 1 essentially
told us that z has to be 1 or negative 1. Statement 2 tells us, z has
to be greater than 0. So if you use both the
conditions combined, it forces us to say, well, then z has
to be equal to positive 1. Because negative 1 is
not greater than 0. So both statements combined
are sufficient to answer this question. Next problem. 127. OK, so they've written
this thing. They write, s is equal
2/n, all of that over 1/x plus 2/3x. And they say in the expression
above, if xn does not equal 0-- so essentially saying that
neither x nor n is 0. What is the value of s? s is equal to what? So even before looking at the
statements, I just want to simplify this. Just because it's too
complex right now. So let's see. This is equal to 2/n
over-- let's see. You have a common
denominator 3x. Let's see. This is 3 over 3x. That's the same thing as 1/x. Plus 2. So that equals 2/n over
5/3x, which is equal to 2/n times 3x/5. Which is equal to 6x/5n. That's a much more pleasing
thing to look at and try to get your brain around. So statement 1 tells us,
x is equal to 2n. So if x is equal to 2n, let's
just substitute that into this statement for s. Then, s would be equal to 6
times x, which we know is equal to 2n. Divided by 5n. n's cancel out. So it equals 12/5. So statement 1, alone,
is sufficient to answer this question. Statement 2 tells us that
n is equal to 1/2. Well, that's fairly useless. Because this is what we
simplified s down to. If we put 1/2 here, then
we get s is equal to 6x over 5 times 1/2. so that's 5/2. I can simplify this more, but we
still don't know what x is. You can't cancel the x's
out or anything. This, alone, doesn't
help you at all. So the answer is statement
1, alone, is sufficient. And statement 2, by itself,
is fairly useless. Problem 128. If x is an integer, is x times--
I think that says the absolute-- it's kind
of strange to look at it like that. But they're saying, is x times
the absolute value of x less than 2x? And they tell us that
x is an integer. OK, statement 1 tells us
that x is less than 0. So how does that help us? If x is less than 0, then this
on the right-hand side is going to be less than 0. And this will also
be less than 0. Because you'll have
a negative number times a positive number. They'll both be less than 0. If we make x is equal to
negative 1, then this would not-- then this will
not be true. Because you'll have negative
1 times 1. So you'll have 1 is less
than negative 2. Sorry, you'd have negative 1
times 1, which is negative 1. Which is less than 2,
which is not true. It's not less than 2. Or, if x was less than negative
2, if x was negative 3, then you would have
negative 3 times 3. Because the absolute value
of negative 3. So you would have minus 9 is
less than minus 3 times 2. Minus 6. This is true. So just this condition, alone,
doesn't get us there. Because I can still pick an x
that meets this condition. And depending on whether I make
that x greater than or less than negative 2, I can
make this true or not. So statement 1, by itself,
isn't enough. Statement 2 tells us, x
is equal to minus 10. Well, this is an easy one. We can just test it. So if we have minus 10 times
the absolute value of minus 10, and we're going to test
whether that's less than 2 times minus 10. So the absolute value of--
So this is minus 10 times positive 10. So it's minus 100, which
is less than minus 20. Which is completely true. So statement 2, alone,
is sufficient to answer this question. And I'm almost out of
time, so I'll see you in the next video.