**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 a^{p} b^{q} c^{r }

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 = 2^{1}3^{3}

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 = 2^{3}3^{1}5^{1}

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

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