Competition problem — Number theory

Library / /320 of 520

Number theory Difficulty 5.7 AIME, harder Prove it

5. Given $n \geqslant 2, n \in \mathbf{Z}_{+}, d_{1}, d_{2}, \cdots, d_{r}$ are all positive divisors of $n$ that are less than $n$ . If the least common multiple of $d_{1}, d_{2}, \cdots, d_{r}$ is not equal to $n$ , find $n$ .

Solution

5. The required $n$ is all prime numbers.

When $n$ is a prime number, 1 is the only positive divisor of $n$ that is less than $n$ , and $1 \neq n$ , so prime numbers satisfy the condition.

When $n$ is a composite number, obviously, the least common multiple of $d_{1}, d_{2}, \cdots, d_{r}$ is $n$ .

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