OG10 #226 S, U,V

oquiella wrote:If s, u, and v are positive integers and 2^s=2^u+ 2^v, which of the following must be true?


I. s=u
II. u does not = v
III. s>v


A. None
B. I only
C. II only
D. III only
E. II and III

quick approach?

Ok, I suspect you have figured this out by now. So in case you or anyone else is still interested here's a way bang this one out.

Notice that the question asks which "must be true."

So if we can come up with any example that does not fit a statement, then we can eliminate that statement.

I. s=u

This one is easy to eliminate. Clearly if v is a positive integer then s > u. So I is out.

II. u does not = v

We can do 2^2 = 2^1 + 2^1. So u can = v. So II is out.

III. s>v

Let's make u as low as possible. Given that s, u, and v are positive, the lowest possible value for u is 1.

So we have 2^s = 2 + 2^v. All the numbers are positive here. So without even doing any math really, we can tell that 2^s has to be greater than 2^v. So s has to be greater than v.

III is the only one that must be true and the correct answer is D.