## Task B-3.1.

Determine all three-digit numbers that when divided by two give a remainder of 1, when divided by three give a remainder of 2, and when divided by four give a remainder of 3. How many such numbers are there?
## Solution.

Let $n$ be the desired number.

According to the conditions of the problem, we can write $n=2a+1, n=3b+2, n=4c+3$.

1 point

If we add one to the desired number $n$, $n+1$ will obviously be divisible by 2, 3, and 4, that is, it will be divisible by 12.

1 point

Therefore, $n+1$ is of the form $12k$, or rather, $n$ is of the form $12k-1$.

1 point

Since $n$ is a three-digit number, it holds that $100 \leqslant 12k-1 < 1000$ where $9 \leqslant k \leqslant 83$. 1 point

The desired numbers are $107, 119, \ldots, 995$.

1 point

There are 75 of them.