12. The greatest power of 7 which is a factor of 50 ! is $7^{k}(n!=1 \times 2 \times 3 \times 4 \times \ldots \times(n-1) \times n)$. What is $k$ ?
A 4
B 5
C 6
D 7
E 8
SolUtion
$\mathbf{E}$
$50!=1 \times 2 \times \cdots \times 50$. Because the seven numbers $7,14,21,28,35,42$ and 49 are divisible by 7 , they each contribute 1 to the power of 7 which is a factor 50 ! In addition, because 49 is divisible by $7^{2}$, it contributes an additional power of 7 .
Therefore the highest power of 7 that divides 50 ! is $7+1=8$.