Track / Stage 5 / 363 of 400 #963 of 2000

Problem 963

AIME late

Number theory Difficulty 5.9 Prove it

4. Let $p$ and $q, (p \neq q)$ be prime numbers such that the product $pq$ is a twenty-digit number. Prove that in the representation of the product $pq$ , at least one digit appears at least three times!

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solution. Let the product $pq$ be a twenty-digit number and let no digit in its representation appear more than twice. However, since the representation has 20 digits, each digit must appear exactly twice. Therefore, the sum of the digits of the number $pq$ is

$2(1+2+3+4+5+6+7+8+9)=90$

which means it is divisible by 9. However, the only prime divisors of $pq$ are $p$ and $q$ , from which it follows that $p=q=3$ . The latter contradicts the condition $p \neq q$ , which implies that in the representation of $pq$ , at least one digit appears at least three times.

Note. The conclusion $p=q=3$ also contradicts the condition that $pq$ is a twenty-digit number.

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