Theorem 1.3 : Let p be a prime number. If p divides `a^2` , then p divides a, where a is a positive integer.

Let p be a prime number such that 3

Let p be a prime number and a be a positive integer.If p divides a^(2); then p divides a.

Prove that the square of any prime number p ge 5 , when divided by 12, gives 1 as remainder.

Let p, q be prime numbers such that non n^(3pq)-n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.

If P is a prime number and P divides ab i.e., P|(ab) where a and b are integers, then

The contra-positive of the statement, "If x is a prime number and x divides ab then x divides a or x divides b", can be symbolically represented using logical connec- tives, on appropriately defined statements p, q, r, s, as

Let P(x) be a polynomial such that p(x)-p'(x)= x^(n) , where n is a positive integer. Then P(0) equals-