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Problem 70

Number theory Difficulty 1.2 Find the answer CEMC Fermat · Canada · 2021

Which of the following integers cannot be written as a product of two integers, each greater than 1?

$6$
$27$
$53$
$39$
$77$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

We note that $6 = 2 \times 3$ and $27 = 3 \times 9$ and $39 = 3 \times 13$ and $77 = 7 \times 11$ , which means that each of 6, 27, 39, and 77 can be written as the product of two integers, each greater than 1.
Thus, 53 must be the integer that cannot be written in this way. We can check that 53 is indeed a prime number.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.