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, 2020

Number theory Difficulty 1.4 Junior Find the answer Canada

Anna thinks of an integer.

It is not a multiple of three.
It is not a perfect square.
The sum of its digits is a prime number.

The integer that Anna is thinking of could be

$12$
$14$
$16$
$21$
$26$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solutions — 2

Solution 1

Since 12 and 21 are multiples of 3 ( $12 = 4 \times 3$ and $21 = 7 \times 3$ ), the answer is not (A) or (D).
16 is a perfect square ( $16 = 4 \times 4$ ) so the answer is not (C).
The sum of the digits of 26 is 8, which is not a prime number, so the answer is not (E).
Since 14 is not a multiple of a three, 14 is not a perfect square, and the sum of the digits of 14 is $1+4=5$ which is prime, then the answer is (B) 14.

Solution 2

12 and 21 are multiples of 3 ( $12 = 4 \times 3$ and $21 = 7 \times 3$ ) so the answer is not (A) or (D).

16 is a perfect square ( $16 = 4 \times 4$ ) so the answer is not (C).

The sum of the digits of 26 is 8, which is not a prime number, so the answer is not (E).

Since 14 is not a multiple of a three, 14 is not a perfect square, and the sum of the digits of 14 is $1+4=5$ which is prime, then the answer is 14, which is choice (B).

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