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Problem 303

Number theory Difficulty 2.4 Multiple choice CEMC Cayley · Canada · 2023

Karim has 23 candies. He eats $n$ candies and divides the remaining
candies equally among his three children so that each child gets an
integer number of candies. Which of the following is not a possible
value of $n$ ?

Pick one

Official solution

After Karim eats $n$ candies,
he has $23-n$ candies
remaining.

Since he divides these candies equally among his three children, the
integer $23-n$ must be a multiple of
3.

If $n = 2, 5, 11, 14$ , we obtain
$23 - n = 21, 18, 12, 9$ , each of
which is a multiple of 3.

If $n = 9$ , we obtain $23 - n = 14$ , which is not a multiple of
3.

Therefore, $n$ cannot equal 9.

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