Mock AIME 1 Pre 2005 Problems/Problem 9

Problem

$p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and \[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1\] Compute $pqr$.

Solution

\begin{align*} \frac {1}{p} + \frac {1}{q} + \frac {1}{r} + \frac {360}{pqr} & = 1 \\ pq + pr + qr + 360 & =  pqr \\ 360 & =  pqr - pq - pr - qr \\  & =  (p - 1)(q - 1)(r - 1) - (p + q + r) + 1 \\  & =  (p - 1)(q - 1)(r - 1) - 25 \\ 385 & =  (p - 1)(q - 1)(r - 1) \\ \end{align*}

From here, you can factor $385$ as $5 \cdot 7 \cdot 11$, giving corresponding values of $6, 8,$ and $12$. The answer is $6 \cdot 8 \cdot 12=576$.

See also

Mock AIME 1 Pre 2005 (Problems, Source)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Invalid username
Login to AoPS