For every positive even integer n, the function h(n) is defined to be the product of all the even integers fro

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100)+1, then p is


A. between 2 and 10
B. between 10 and 20
C. between 20 and 30
D. between 30 and 40
E. greater than 40 (correct answer)

someone was explaining this following...

"first consider the prime factors of h(100). According to the given function,
h(100) = 2*4*6*8*...*100

By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50). "

However, I have no idea why you can convert to that form. If it were 2*(1*2*3*4*.....50), then i would understand even though I can go any further from there....

Please someone help me out to get this. Thank you so much!!

You are confusing this with the distributive property. If the problem were this: 2 + 4 + 6 + ... + 100 then you could factor out a 2 from each term to get 2(1 + 2 + 3 + ... + 50). But it is multiplication, not addition. Try to follow these steps:

2 * 4 * 6 * 8 * ... * 100
(2 * 1) * (2 * 2) * (2 * 3) * (2 * 4) * ... * (2 * 50)
2 * 1 * 2 * 2 * 2 * 3 * 2 * 4 * ... * 2 * 50
2 * 2 * 2 * ... 2 * 1 * 2 * 3 * 4 * ... * 50
2^50 * 1 * 2 * 3 * 4 * ... * 50

EDIT: because of order of operations, only the 50 is the exponent. The above id equivalent to:

(2^50) * (1 * 2 * 3 * 4 * ... * 50)

Can you see how to proceed from here? Hint: h(100) is divisible by ALL integers from 2 to 50. What does this imply about h(100) + 1?