Question

Find the number of ways of expressing \[216\] as a product of two Factors

Answer 1

Hint: Prime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are called composite numbers.

Complete step-by-step answer:
Write a prime factorization of a given number.
i.e. write the given number as a product of prime numbers.
 $ 216 = {2^3} \times {3^3} $ . . . (1)
 $ 216 $ is not a perfect square.
The numbers of ways of expressing any non-perfect square as a product of two Factors is given by
 $ \dfrac{1}{2} $ (number of its Factors) . . . (2)
Number of factors:
Let us say, $ x $ is some number which can be written as
 $\Rightarrow x = {a^p}{b^q}{c^r} $
Where, $ a,b $ and $ c $ are the prime factors of $ x. $ And $ p,q $ and $ r $ are the respective powers.
Then, the number of factors of $ x = (p + 1)(q + 1)(r + 1) $
Therefore, from equation (1), we can write
Numbers of factors of $ 216 = (3 + 1)(3 + 1) $
 $ = 16 $
Therefore, from equation (2), we can write
The number of ways of expressing 216 as a product of two factors is
 $ = \dfrac{1}{2} \times 16 $
 $ = 8 $
Therefore, the numbers of ways of writing $ 216 $ as a product of two factors $ = 8 $

Note: You have to make sure that the number given in the question is not a perfect square before using the said formula. The factors must be prime factors to get the correct answer.