Track / Stage 3 / 192 of 260 #192 of 1964

Problem 192

AMC 10/12, early questions

Number theory Difficulty 3.7 Find the answer

Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

If an integer is divisible by $36$ , it must also be divisible by $9$ since $9$ is a factor of $36$ . It is a well-known fact that, if $N$ is divisible by $9$ , the sum of the digits of $N$ is a multiple of $9$ . Hence, if $N$ contains all the even digits, the sum of the digits would be $0 + 2 + 4 + 6 + 8 = 20$ , which is not divisible by $9$ and thus $36$ .
The next logical try would be $8640$ , which happens to be divisible by $36$ . Thus, $N = 8640 \equiv \boxed{640} \pmod {1000}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.