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.