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 (not equal to) v
III. s > v
A. none
B. I only
C. II only
D. III only
E. II and III
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An open Blog with tentative solutions and discussion of GMAT questions
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1 comment:
In other words, we are looking for a NECESSARY CONDITION (I, II or III) for the equation to be true with positive integer values for {s, u, v}. That's what we get from the question form "which (..) must be true?".
Two strategies were possible for this class of exercises. We might a) analyse the given expression and look for the validity universe on {s,u,v} for the equation 2^s=2^u+2^v and then look for match with the three propositions given. Or b) we can look for consequences of the three given conditions on the equation. It seems that the second approch (b) can be more straightforward, as the number of propositions to be tested is not big. So let's stat the tests:
condition I (s=u) yields 2^u = 2^u + 2^v which, to be true implies that 2^v is zero. This is an impossible condition, as v is a positive integer. Conclusion: condition I doesn't have to be true.
condition II [s (different from)u] wouldn't be a necessary condition if we just find a counter-example, that is a set {u,v} with u=v that satisfies the equation. We have such an immediate counterexample with u=v=1 and s=2, leading to the obvious "truth" 4=2+2. So, condition II is also not necessary.
condition III seems truly necessary, even intuitively. In fact, if it wasn't so (i.e. s<=v) then (2^s - 2^v) = 2^u would have to be zero or negative, which is actually impossible when, as said, u is also a positive integer.
This leads to the conclusion that condition III only is necessary. The correct answer is (D)
v is zero. This is an impossible condition, as v is a positive integer. Conclusion: condition I doesn't have to be true.
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