Track / Stage 4 / 263 of 340 #523 of 1964

Problem 523

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

Variant 9.3.2. Natural numbers $a$ and $b$ are such that $a$ is divisible by $b+1$ and 67 is divisible by $a+b$ .

(a) (1 point) Indicate any possible value of $a$ .

(b) (3 points) What can $b$ be? Indicate all possible options.

Your answer

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

Official solution

Answer: (a) any of the numbers $34,51,64,66$ . (b) $1,3,16,33$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.