Question

When a prime number divides the product of two natural numbers, it ensures that:

a) It divides both the numbers
b) It doesn't divide any of the numbers
c) It divides at least one of the numbers
d) It divides the sum of the numbers

Answer 1

A prime number dividing the product of two natural numbers indicates that it must be a factor of at least one of those numbers.

Step-by-step explanation:

When a prime number divides the product of two natural numbers, it ensures that it divides at least one of the numbers. This is based on a principle known as the Fundamental Theorem of Arithmetic, which states that any integer greater than 1 is either a prime number or can be factored into a unique combination of prime numbers (its prime factorization). Therefore, if a prime number can exactly divide the product of two numbers without leaving a remainder, then that prime number must be a factor of at least one of those two numbers.