factor of h (100) + 1

sanju09 wrote: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

E

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
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This is tricky.
h(100)= 2*4*6*.........*100
take out 2 from each even number
h(100) becomes 2^50 * (1*2*3*......*50).
h(100) is divisible by all the nos. from 1 to 50 which include prime nos. as well.
so h(100)+1 will not be divisible by any of the number from 1 to 50.
Therefore, ans. prime no. greater than 50.
Option E is correct.

shanconnoisseur wrote: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
===================================

This is tricky.
h(100)= 2*4*6*.........*100
take out 2 from each even number
h(100) becomes 2^50 * (1*2*3*......*50).
h(100) is divisible by all the nos. from 1 to 50 which include prime nos. as well.
so h(100)+1 will not be divisible by any of the number from 1 to 50.
Therefore, ans. prime no. greater than 50.
Option E is correct.

beautiful