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Problem 109

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 33 as when it is divided
by 44?

Pick one

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Official solutions — 2

Solution 1

Since 9=3×39=3\times3, then 99 leaves remainder 00 when divided by 33. Since 9=2×4+19=2\times4+1, then 99 leaves remainder 11 when divided by 44. The remainders upon division by 33 and also division by 44 for each of the other given numbers are
in the table that follows.

Integer
99
1111
1010
1313
88

Remainder when divided by 33
00
22
11
11
22

Remainder when divided by 44
11
33
22
11
00

Of the given integers, 1313 has
the same remainder when it is divided by 33 as when it is divided by 44.

Solution 2

Since 9=3×39=3\times3, then 99 leaves remainder 00 when divided by 33. Since 9=2×4+19=2\times4+1, then 99 leaves remainder 11 when divided by 44. The remainders upon division by 33 and also division by 44 for each of the other given numbers are
in the table that follows.

Integer
99
1111
1010
1313
88

Remainder when divided by 33
00
22
11
11
22

Remainder when divided by 44
11
33
22
11
00

Of the given integers, 1313 has
the same remainder when it is divided by 33 as when it is divided by 44.

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