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Lesson: Intermediate Problem Solving - 13t01

The "Must Be" Question

[Page 13 of 27]
On questions that ask "Which of the following must be true?" or "Which of the following must be false?", use Picking Numbers to eliminate answer choices.

The key to these types of problems is to look for a number that will disqualify the answer choice:

  • If the question asks for the answer choice that must always be true, try to find a number that will make the answer choices false.
  • If the question asks for the answer choice that must always be false, try to find numbers that will make the answer choices true.
Review the following question. For each answer choice, pick numbers that will allow you to eliminate the choice. To see the numbers we've picked and the result, CLICK ON EACH ANSWER CHOICE. Click Continue when you are done.

If m and n are integers such that mn = 500, which of the following statements must be true?

m + n > 0
Either m or n is a multiple of 10.
m + n is odd
If m is odd, then n is even.
If m is even, then n is odd


The only choice for which we were unable to find an exception is Choice (D), so it's the correct answer.

It would have been easy to pick numbers to make the statements in the answer choices true. But, if we'd done that, we wouldn't have been able to eliminate any choices! The key for "must be true" questions is to find numbers that can make four of the five choices not true.

And one more note: Because this is a "Which of the following" question you should have started with Choice E to reinforce your test day habit. But you couldn't have stopped when you got to Choice (D) and found that you couldn't find an exception. That's because this is a "must be true" question. You can prove that something cannot be true by counterexample, which is what we did when we eliminated choices above. But you can't prove that it must be true by picking numbers, because it might be that you simply hadn't picked the correct numbers! Eliminating down to one choice remaining was the only way to apply picking numbers to this problem with certainty.

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