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Problem 221

Number theory Difficulty 1.6 Multiple choice CEMC Cayley · Canada · 2026

Let $s$ be the sum of the
digits of a positive two-digit integer $n$ . Which of the following is a possible
value of $n-s$ ?

Pick one

Official solution

If the positive two-digit integer $n$ has tens digit $x$ and units digit $y$ , then $n=10x+y$ and $s=x+y$ . Then $n-s=(10x+y)-(x+y)=9x$ , and so $n-s$ is a multiple of $9$ (since $x$ is a positive integer). Of the answers
given, the only multiple of $9$ is
$54$ .

(If for example $n=62$ , then $n-s=62-(6+2)=54$ .)

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