Question

Prove or disprove: If p is a prime and a is an integer, then gcd(p,a)=p if and only if p divides a.

A) True
B) False
C) Depends on the values of p and a
D) Can't be determined

Answer 1

The statement is true; gcd(p, a) = p implies p divides a, and vice versa, because p is a prime and can only be divided by itself and 1.

Step-by-step explanation:

The statement is true. The greatest common divisor (gcd) of a prime number p and an integer a is p if and only if p divides a. The concept of gcd relates to the largest positive integer that divides each of the numbers without leaving a remainder. In this context, since p is a prime number, the only positive integers that can divide it are 1 and p itself. If the gcd of p and a is p, it means that p is a divisor of a.

This is because the gcd cannot be greater than the smallest number, which in this case is p. True. If p is a prime and a is an integer, then gcd(p,a) = p if and only if p divides a. To prove this, we can use the definition of the greatest common divisor (gcd). Conversely, if p divides a, then the gcd must be p as no larger number could be a divisor of p. Therefore, the statement is true by the definition of both prime numbers and the gcd function.