Prove that following statements.
If n is positive integer and 3 divides `n^2` then 3 divides n.

State the negations for the following statements : 2 divides the positive integer a.

Prove that (n!)^2 le n^n. (n!)<(2n)! for all positive integers n.

Find counter-examples to disprove the following statement: n^2 -n + 41 is a prime for all positive integers n

Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

If n is any positive integer , show that 2^(3n +3) -7n - 8 is divisible by 49 .

For every positive integer n, prove that 7^n-3^n is divisible by 4.

Prove that the product of 2n consecutive negative integers is divisible by (2n)! .

Show that (-sqrt(-1))^(4n+3)=i , where n is a positive integer.

If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is

Prove that n!+1 is not divisible by any integer between 2 and n.