Problem 683

3. If a number is divisible by 2, 3, and 7, then it must be divisible by $2 \cdot 3 \cdot 7=42$ and vice versa.

2 points

Since $184952=42 \cdot 4403+26$ and 2 points

the largest three-digit number $999=42 \cdot 23+33$, 2 points

then the sum $184952+999=42 \cdot 4403+26+42 \cdot 23+33=$

$$\begin{aligned} & =42 \cdot 4403+42 \cdot 23+42+17= \\ & =42 \cdot(4403+23+1)+17 \end{aligned}$$

We see that this sum is not divisible by 42 because the remainder when divided by 42 is 17. 2 points We conclude that 999 - $17=982$ is the largest three-digit number that needs to be added to 184952 so that their sum is divisible by 42.