<p>These appear on every SAT and I often just do guess and check which I realize is a time-consuming approach. So, what's the efficient way of solving these? Here's an example:</p>
<p>"If a and b are positive integers, which of the following must be a positive odd integer?"
A. a + b
B. a - b
C. 2a + b
D. 2a - b
E. (a + b)/2</p>
<p>None are correct, actually. A through D are false because we can let a and b be even, and E is false because we can find a and b such that a+b is divisible by 4.</p>
<p>Since you’re working modulo 2, you only need to test with 0 and 1, which I find to be a good strategy. If you’re working with multiples of 3, try 0, 1, 2.</p>
<p>^SORRY, jeez, I meant to put “If a and b are positive odd integers”.</p>
<p>Anyway, so you’re saying the only way is to guess and check?</p>
<p>a = 2m+1
b = 2n+1</p>
<p>(a+b)=2(m+n+1). Even
a-b=2(m-n) even
2a+b = 4m+2n+3 odd
2a-b = 4m-2n+1 may not be odd if</p>
<p>Hmm… (A+b)/2 is also odd.</p>
<p>Is the question correct?</p>
<p>Yes, you should type the question as is. Missing important words like “odd” confuse us.</p>
<p>A good strategy is to try finding counterexamples. A and B are clearly incorrect (those are always even). D yields a number that is always odd, but it is not always positive, since b can be arbitrarily large. E is false since we can let a=1, b=3. The correct answer is C.</p>
<p>The easiest way is to guess and check. Just pick two small numbers, like A=3 and B=1, and then you can blow right through the answer choices, and see which one’s odd.</p>
<p>Exactly, Plug in for all/most variable questions.</p>
<p>Backsolve and Elimination comes on other math questions.</p>