Examples

The following example illustrates the concept of reducibility. Consider three problems, A, B, and C:

• A=Prime(n): The problem of determining whether or not n is a prime number.
• B=Numberfactor(n): The problem of counting the number of distinct primes that divide n.
• C=Smallestfactor(n): The problem of finding the smallest integer such that x divides n.

In this example A C, and B C. Why?

To find if n is a prime, we attempt to divide it by 2, 3, 5, etc. The smallest integer that divides n also answers A.

Thus the solution of Smallestfactor(n) tells us that n is not a prime.

To see how B C we need a simple algorithm that counts the number of distinct divisors of n using C.