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

I) s = u
II) u (not equal) v
III) s>v

a)none
b)I only
c)II only
d)III only
e) II and III

The answer is D.

I was really not sure how to approach this problem. I started by eliminating the 2's to get s = u + v.

I was able to test out numbers to eliminate I. For example 3 = 3 + anything will be false.. and we have positive integers.

I was not sure how to eliminate II and also would appreciate some help with III.

Thanks,
Carla

Carla,

Whenever a problem states, "which of the following must be true", simply try to disprove each of the following.

You have shown how to disprove (I).

To disprove (II), try using numbers where u and v are equal; u = 0.5, v = 0.5:

2S = 2(0.5) + 2(0.5)
2S = 1 + 1
2S = 2
S = 1

Therefore, u (not equal) v does not have to be true.

(III) cannot be disproven. This is because the question states that s, v, and u are all positive integers.

Hope this helps!

That did help - I appreciate the feedback very much!
-Carla

Quick citation note: This is from the Official Guide for GMAT Quantitative Review. For copyright reasons, we must cite full source name.

Good explanation, GMAT 5/18, thanks for the assist!

Carla -

You're right that it simplifies to s = u + v where s,u,v are positive integers.

1) since u and v are both positive integers, it doesn't have to be the case that s = u. In fact it can't be the case that s = u.

2) There is no restriction that keeps disallows u = v. For example s=4, u=2, v=2 works.

3) If u and v are positive integers, then s must be greater than v. This is because both u and v are positive integers and s is there sum. So s must be larger than both u and v.

So only 3 must be true. The key to this problem is not to lose site that s,u,v are all positive integers. It gets down dramatically on the possibilities that you need to consider, e.g. zero, negative numbers and non-integers.

cheers,
Jeff

Good work, guys (and by "guys," I mean guys and gals)!