Question 104
Given
$$(x^2+5x-19)^{50}=a_0+a_1x+\cdots+a_{100}x^{100},$$
$$M=a_0-a_1+4a_2-9a_3+16a_4-81a_5+64a_6-729a_7+\cdots+2^{98}a_{98}-3^{98}a_{99}+2^{100}a_{100}.$$
The coefficient of $a_{2k}$ in $M$ is $2^{2k}$ and that of $a_{2k+1}$ is $-3^{2k}$. What is the greatest power of 5 dividing $M$?
(A)$5^0$
(B)$5^1$
(C)$5^5$
(D)$5^{50}$
(E)$5^{100}$
$$(x^2+5x-19)^{50}=a_0+a_1x+\cdots+a_{100}x^{100},$$
$$M=a_0-a_1+4a_2-9a_3+16a_4-81a_5+64a_6-729a_7+\cdots+2^{98}a_{98}-3^{98}a_{99}+2^{100}a_{100}.$$
The coefficient of $a_{2k}$ in $M$ is $2^{2k}$ and that of $a_{2k+1}$ is $-3^{2k}$. What is the greatest power of 5 dividing $M$?
(A)$5^0$
(B)$5^1$
(C)$5^5$
(D)$5^{50}$
(E)$5^{100}$