Intermediate digits of 34! [duplicate]

contest-math elementary-number-theory decimal-expansion factorial number-systems

Solution 1:

HINT: The number is a multiple of $9$ and a multiple of $11$.

Solution 2:

You know that $34!$ is divisible by $9$, so, because the sum of all other digits is $141$, you know that $c+d=3$ or $c+d=12$.

Now do the same for divisibility by $11$ (remember the alternating sums criterion?).

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