Track / Stage 5 / 91 of 400 #691 of 1964

Problem 691

AIME late

Number theory Difficulty 5.3 Find the answer

6. (15 points) For a natural number $N$ , if at least six numbers in the nine natural numbers from $1 \sim 9$ can be factors of $N$ , then $N$ is called a “Six-Union Number”. Among the natural numbers greater than 2000, the smallest “Six-Union Number” is $\qquad$

Your answer

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

Official solution

【Answer】Solution: According to the problem, this六合number must have a factor of 1, and it must be an even number. Among these numbers, the smallest is 2016, which meets the conditions of a六合number. Therefore, the answer to this question is 2016.

Note: "六合number" is a term that doesn't have a direct translation in English. It might refer to a specific type of number in a particular context. For the purpose of this translation, I've kept it as "六合number" to maintain the original meaning. If you have a specific term or context for "六合number," please let me know, and I can adjust the translation accordingly.

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