<s>[INST] $17!=3556xy428096000,$then $(x+y)$ equals?(without using a calculator) $17!=3556xy428096000,$then $(x+y)$ equals?
a)$15$ b)$6$ c)$12$ d)$13$
With help of calculator $(x+y)$ can be easily calculated as $15$.But without a calculator,I can only conclude that the sum of digits $(3+5+5+6+x+y+4+2+8+9+6)$ is a multiple of $3$, & excluding $(x+y)$ the sum of the digits $(3+5+5+6+4+2+8+9+6) =48$.Thus excluding option d) the ans can be any of a),b) or c).
Please help me to find the exact answer.
Thank you 
 [/INST] Testing divisibility by 9 gives
$$
48 + x + y \equiv 0 \pmod {9}.
$$
Testing divisibility by 11 gives
$$
12 - x + y \equiv 0 \pmod {11}
$$
So $x - y \equiv 1 \pmod {11}$ and $x + y \equiv 6 \pmod{9}$.
Since $x, y$ are both digits, $x = 1 + y$, and $x + y$ is odd.
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