2002 Math OSK, Number 3: Raising Rules

Note first that

$44 = 2^2 \times 11$ .

Next, we evaluate

$44^{44} = 2^{88} \times 11^{44}$

$44^{44} = {(2^3)}^{29} \times 2 \times 11^{44}$

$44^{44} = {8}^{29} \times 2 \times 11^{44}$

From this we see that $44^{44}$ is divisible by $8^{29}$ . Thus, $n = 29$ .

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