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Number theory Difficulty 5.2 AIME, harder Prove it United States

Problem:

Positive integers $a$ , $b$ , and $c$ have the property that $a^{b}$ , $b^{c}$ , and $c^{a}$ end in $4$ , $2$ , and $9$ , respectively. Compute the minimum possible value of $a+b+c$ .

Solution

Solution:

This minimum is attained when $(a, b, c) = (2, 2, 13)$ . To show that we cannot do better, observe that $a$ must be even, so $c$ ends in $3$ or $7$ . If $c \geq 13$ , since $a$ and $b$ are even, it's clear $(2, 2, 13)$ is optimal. Otherwise, $c = 3$ or $c = 7$ , in which case $b^{c}$ can end in $2$ only when $b$ ends in $8$ . However, no eighth power ends in $4$ , so we would need $b \geq 18$ (and $a \geq 2$ ), which makes the sum $2 + 18 + 3 = 23$ larger than $17$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.