Consider the positive integers p, q, and r such that 4p = 4q + 4r. Which of the following statements must hold true?

I. $p = q$

II. $q = r$

III. $p > r$

Correct answer marked below

None

I only

II only

III only

Correct

II and III

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Explanation

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Official Explanation

Official

Given that p, q, and r are positive integers and the equation 4p = 4q + 4r holds, we can simplify this to

p = q + r

.

Since p, q, and r are all positive integers, it follows that p is greater than either q or r, making statement I false and statement III true.…

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What this question tests

Your ability to manipulate linear and quadratic expressions, substitute unknowns, and solve equations under time pressure.

Losing a sign when moving terms across the equals sign, or dividing both sides by a variable that could equal zero.