If $ a_1 a_2a_3 ...a_{20} = 2^x * y! $ Then what is the value of (x+y)?

Question

Let us consider a series with $ a_1 = 2012 $ and $ a_n = \frac{n}{a_{n-1}} $ . If $ a_1 a_2a_3 ...a_{20} = 2^x * y! $ Then what is the value of (x+y) ?

Answer 1

HINT:

As $a_na_{n-1}=n, a_{2r+1}a_{2r+2}=2r+2=2(r+1) $

$$\prod_{1\le r\le 20}a_r=\prod_{0\le r\le9}a_{2r+1}a_{2r+2}=2^{10}\prod_{0\le r\le9}(r+1)=2^{10}(10)!$$

Now, we can use this, to find the highest power of $2$ in $10!$ and also the highest power of other primes $<10$ which constitute $y$