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Smallest Prime Number

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by xs2infa » Sat Jul 16, 2011 12:53 am
For every positive even integer n, F(n) is defined as the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of F(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

Can the experts let me know the correct answer along with the explanation?
Thanks
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by Anurag@Gurome » Sat Jul 16, 2011 3:10 am
Refer to this post : https://www.beatthegmat.com/ps-problem-s ... tml#322262
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by xs2infa » Sat Jul 16, 2011 3:30 am
Thanks Anurag.
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by sanju09 » Sat Jul 16, 2011 4:42 am
xs2infa wrote:For every positive even integer n, F(n) is defined as the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of F(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

Can the experts let me know the correct answer along with the explanation?
Thanks

By definition, F (100) = 2 × 4 × 6 × ...× 100 = 2^50 × (1 × 2 × 3 × ...× 50), hence F (100) includes all prime numbers up to 50 as its factor.

If we divide the value F (100) + 1 by any of the prime number in the list (1 × 2 × 3 × ...× 50), then the remainder would be 1 in each case. Hence no prime number less than 50 can be a factor of the value F (100) + 1. The best answer choice available here is E
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