If $a, b$ and $c$ are distinct positive integers such that $a b c=16$, then the largest possible value of $a^{b}-b^{c}+c^{a}$ is
(A) 253
(B) 63
(C) 249
(D) 263
(E) 259
If $a, b$ and $c$ are distinct positive integers such that $a b c=16$, then the largest possible value of $a^{b}-b^{c}+c^{a}$ is
(A) 253
(B) 63
(C) 249
(D) 263
(E) 259

## Solution

If $a, b$ and $c$ are distinct then the correct factorization is $16=1 \times 2 \times 8$. Since $a, b$ and $c$ must be some permutation of 1,2 and 8 there are exactly six possibilities which give the values $-247,-61,65,249$, 263, and 63. Of these, $8^{1}-1^{2}+2^{8}$ or 263 is the largest.

ANSWER: (D)