Number Properties – Product of factors

What is the product of all positive factors of 18?

A. 324

B. 5832

C. 39

D. 1521

E. 3042

The correct answer is 5832. Choice B.

Let us first focus on how to find the answer methodically and then understand why the method works.

The How?

Step 1: Find the number of factors for 18.

Step 1a. Express 18 as a product of its prime factors. 18 = 2 * 3^2

Step 1b. Increment the powers of each of the prime factors. The power of 2 is 1. The power of 3 is 2. After incrementing, we get (1 + 1) and (2 + 1)

Step 1c. Number of factors = Product of the incremented numbers = (1 + 1)(2 + 1) = 2 * 3 = 6

Step 2: The product of all factors of 18 = 18^(number of factors/2)

The product = 18^(6/2) = 18^3 = 5832.

The Why?

The factors of 18 are 1, 2, 3, 6, 9 and 18.

Product of the factors = 1*2*3*6*9*18.

We can rewrite this as (1*18)*(2*9)*(3*6) = 18 * 18 * 18.

So, if a positive integer ‘n’ has ‘x’ factors, then the product of all its factors = n^(x/2).

What if the number has odd number of factors?

Try the method for a perfect square such as 36 or 16.