For every pos. even int. n, h(n) is defined to be the...

[dwill]

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

a. between 2 and 10
b. between 10 and 20
c. between 20 and 30
d. between 30 and 40
e. greater than 40

Can you present the steps to solving this problem?

 

[Count Iblis]

Re: function problem

For every positive even integer n, the function h(n) is defined to be the product of all 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. between 40 and 50
f. greater than 50

Can you present the steps to solving this problem?

Answer f is correct.

Note that:

\(\displaystyle \L h(100) =\prod_{j=1}^{50}(2j)=2^{50}50!\)

If some prime \(\displaystyle p\) divides \(\displaystyle h(100)+1\) then \(\displaystyle h(100)=-1 \operatorname{mod} p\) but if \(\displaystyle p\leq 50\) then \(\displaystyle h(100)= 0 \operatorname{mod} p\)