Competition problem — Number theory

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Number theory Difficulty 5.0 AIME, harder Find the answer Netherlands

For a three-digit positive integer, we can multiply the three digits together. We call the result the digit product of that number. For example, $123$ has digit product $1 \times 2 \times 3 = 6$ and $524$ has digit product $5 \times 2 \times 4 = 40$ . A number cannot begin with the digit $0$ .
Determine the three-digit number that equals exactly five times its own digit product.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

$175$

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