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Problem 193

Number theory Difficulty 1.3 Multiple choice CEMC Pascal · Canada · 2026

Which of the following integers has the same remainder when it is
divided by $3$ as when it is divided
by $4$ ?

Pick one

Official solutions — 2

Solution 1

Since $9=3\times3$ , then $9$ leaves remainder $0$ when divided by $3$ . Since $9=2\times4+1$ , then $9$ leaves remainder $1$ when divided by $4$ . The remainders upon division by $3$ and also division by $4$ for each of the other given numbers are
in the table that follows.

Integer	$9$	$11$	$10$	$13$	$8$
Remainder when divided by $3$	$0$	$2$	$1$	$1$	$2$
Remainder when divided by $4$	$1$	$3$	$2$	$1$	$0$

Of the given integers, $13$ has
the same remainder when it is divided by $3$ as when it is divided by $4$ .

Solution 2

Integer	$9$	$11$	$10$	$13$	$8$
Remainder when divided by $3$	$0$	$2$	$1$	$1$	$2$
Remainder when divided by $4$	$1$	$3$	$2$	$1$	$0$

Of the given integers, $13$ has
the same remainder when it is divided by $3$ as when it is divided by $4$ .

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