whether prime number p ia a factor of n!

[mariah]

is the prime number p a factor of n!, where n is a nonnegative integer.

1. Prime number p divides n! + (n+2)!.
2. The prime number p divides (n+2)!/n!.

[spoiler]oa c[/spoiler]

[manpsingh87]

is the prime number p a factor of n!, where n is a nonnegative integer.

1. Prime number p divides n! + (n+2)!.
2. The prime number p divides (n+2)!/n!.

[spoiler]oa c[/spoiler]

1) n!+(n+2)!; n!(1+(n+1)(n+2)) now consider n=1, and p=7; expression n!+(n+2)! is divisible by 7 but 1! is not divisible by 7, now consider n=5 and p=3; expression n!+(n+2)! is divisible by 3 and also 5! is also divisible 3, hence 1 alone is not sufficient to answer the question.

2) (n+2)!/n!; (n+2)(n+1); now consider n=1 p=3; now (n+2)!/n! is divisible by 3 but 1! is not divisible by 3, now consider n=5 and p=3; (n+2)!/n! is divisible by 3 and also 5! is also divisible by 3 hence 2 alone is not sufficient to answer the question.

now consider 1 and 2 together we have;
(n+2)!/n! is divisible by p; therefore (n+2)!/n! will be a multiple of p, hence can be written as,
(n+2)!/n!=pk, where k is an integer;
(n+2)!= pk * n!;----------------------a)
now expression 1 we have; n!+(n+2)!; from a) we have; n!+pk*n!; n!(1+pk); now 1+pk will always give 1 as a remainder when divide by p; therefore for expression n!(1+pk) to become divisible by p n! should be divisible by p..!!

hence C