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Berkeley Math Circle Monthly Contest 4

Number theory Difficulty 5.1 AIME, harder Prove it United States

Problem:

Determine, with proof, whether or not there exist positive integers $a$ , $b$ , and $c$ such that
$a b + b c = a c \quad \text{and} \quad a b c = 10! .$

Solution

Solution:

The answer is no. Note that $10!$ has exactly one prime factor of $7$ . Therefore, exactly one of $a$ , $b$ , and $c$ is divisible by $7$ . If $7$ divides $b$ (we write this as $7 \mid b$ ), then $7 \mid a b + b c$ but $7 \nmid a c$ , so the equation cannot hold. Likewise, if $7 \mid a$ , then $7 \mid a b$ , $7 \mid a c$ but $7 \nmid b c$ , and if $7 \mid c$ , then $7 \mid b c$ , $7 \mid a c$ but $7 \nmid a b$ . So there are no solutions.

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