Track / Stage 4 / 195 of 340 #455 of 2000

Problem 455

AMC 12 late, AIME early

Number theory Difficulty 4.8 Find the answer

20. Find the smallest natural number divisible by 72, in the representation of which all digits from 1 to 9 appear.

Your answer

Official solution

20. The desired number must be divisible by 72, which means it must be divisible by 9 and 8; $1+2+\ldots+9=45$ , which is divisible by 9. For the number to be divisible by 8, it must end with three digits that form a three-digit number divisible by 8. Considering all this, the digits should be arranged so that the smallest ones are to the left. Such a number is 123457968.

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