SAT Math Multiple Choice Practice Question 393: Answer and Explanation
19. If r and s are positive integers and , which of the following must be true?
I. s is odd
II. r is even
is an integer
A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III
Correct Answer: D
D Think carefully about the given information and what it implies, then try to find counterexamples to disprove the given statements. For instance, try to disprove statement I by showing that s can be even. Imagine :
This doesn't work because r must be an integer. Why didn't it work? Because 2r must be even, but if s is even, then s + 1 must be odd and cannot equal an even number, so s must always be odd and statement I is true. (Eliminate choice (B).)
Since 1 is an integer, we've proven that r is not necessarily even, so II is false. (Eliminate choices (C) and (E).)
Since we still have two choices remaining, we have to check ugly old statement III. Try the values we used before. If and , then , which is an integer. But is it always an integer? Plugging in more examples can't prove that it will ALWAYS be an integer, because we can never test all possible solutions. We can prove it easily with algebra, though. Since :
Since 2 is an integer, statement III is necessarily true.