Product of Two factors
To find the number as the products of two factors, use the following steps :
Step1: Write Prime factorisation of given number i.e. convert the number in the form ap bq cr
where a ,b,c are prime numbers and the p,q,r are natural numbers as their respective powers.
Step 2:Find Number of factors which can be expressed as( p+1)(q+1)(r+1).
Step 3: Number of ways to express the number as a product of two numbers is exactly half its number of factors i.e.½ *(p+1)(q+1)(r+1).
Let’s have an example on this :
Example 1: In how many ways can you express 54 as a product of two of its factors?
Solution: We will do the above problem step by step:
Step 1: Prime factorization of 54 i.e. we write 54 = 2133
Step 2: Number of factors of 54 will be (1+1)(3+1) = 2 x 4= 8
Step 3: Hence number of ways to express 54 as a product of two numbers is exactly half its number of factors i.e. ½ *8 = 4 ways.
In fact we can list these 4 ways as well
Factors of 54 are 1,2,3,6,9, 18,27,54.
Now it is very simple to find the factors from 1 to 9 but it is difficult to find the ones that are greater than 10. So number of ways to express 54 as a product of two of its factors is
1 x 54 = 54
2 x 27 = 54
3 x 18 = 54
6 x 9 = 54
Example 2: In how many ways you can express 120 as a product of two of its factors?
Solution:
Step 1: Prime factorization of 120 i.e. we write 120 = 233151
Step 2: Number of factors of 120 will be (3+1)(1+1)(1+1) = 4 x 2 x2 = 16
Step 3: Hence number of ways to express 120 as a product of two numbers is exactly half its number of factors i.e. ½ *16 = 8
In fact we can list these 8 ways as well
Factors of 120 are 1,2,3,4,5,6,8,10,12,15,18,24,30,40,60,120.
So number of ways to express 120 as a product of two of its factors is
1 x 120 = 120
2 x 60 = 120
3 x 40 = 120
4 x 30 = 120
5 x 24 = 120
6 x 20 = 120
8 x 15 = 120
10 x 12=120
