# Problem 2. (2 points)

A natural number $n$ when divided by 12 gives a remainder $a$ and an incomplete quotient $b$, and when divided by 10, it gives a remainder $b$ and an incomplete quotient $a$. Find $n$.
Answer: 119

## Solution:

From the condition, it follows that $n=12 b+a=10 a+b$, from which $11 b=9 a$. Therefore, $b$ is divisible by 9, and $a$ is divisible by 11. On the other hand, $a$ and $b$ are remainders from division by 12 and 10, respectively, so $0 \leqslant a \leqslant 11$ and $0 \leqslant b \leqslant 9$. We get two options: $a=b=0$ and $n=0$ (which does not fit, since $n$ is a natural number) or $a=11, b=9, n=119$.