If p and q are positive integers greater than… | GMAT Sprint

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PS · Problem Solving

705-805

Divisibility/Multiples/Factors

If $p$ and $q$ are positive integers greater than 1 and $p$ > $q$ , for how many pair(s) of values { $p$ , $q$ } is $p^q$ ≥ $q^p$ ?

Correct answer marked below

None

One

Two

Correct

Three

More than three

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Explanation

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Official Explanation

Official

We need integer pairs (p > q > 1) such that

p^q \ge q^p

.

Test small values for q:

- For q = 2:

- p = 3:

3^2 = 9 \ge 2^3 = 8

✓

- p = 4:

4^2 = 16 \ge 2^3 = 8

✓

- p = 5:

5^2 = 25 < 2^3 = 32

✗

For p ≥ 5, the inequality fails, so only p = 3 and p = 4 work.…

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What this question tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Forgetting that 1 is not prime and that every integer divides 0.