Problem Solving - 6

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

Answer - E

h(n)=2*4*6....100 =2(1*2*3*........50)

h(n) + 1 = 2(1*2*...50) + 1

Prime No ={2,3,........}

h(n) contains multiples of all the primes uptil 50 .... Hence E

OR

Let X = 2*4*6*.....*100

then X contains multiples of all the primes uptil 50
eg.
17 * 2 = 34 is in X.

37 * 2 = 74 is in X

Hence all the primes between 2 - 50 will be a factor of X => that none of them will be a factor of (X+1), hence the smallest prime should be at least greater than 50

Hence E.