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Problem 331

Number theory Difficulty 2.4 Multiple choice CEMC Fermat · Canada · 2022

Suppose that $d$ is an odd
integer and $e$ is an even integer.
How many of the following expressions are equal to an odd integer?

$d + d$
$(e+e)\times d$
$d\times d$
$d\times(e+d)$

Pick one

Official solution

Since $d$ is an odd integer,
then $d + d$ is even and $d \times d$ is odd.

Since $e$ is an even integer, then
$e + e$ is even, which means that
$(e+e) \times d$ is even.

Also, $e + d$ is odd, which means
that $d \times (e+d)$ is odd.

Thus, 2 of the 4 expressions are equal to an odd integer.

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