We all have studied about basic division concepts. In this article, we will further extend this concept to understand factors and multiples.

** ****Factors:**

If an integer x is completely divided by another integer y, then x is said to be a multiple of y; and y is said to be a factor of x.

Let us take an example to understand it better.

**Example:**

12 is a multiple of 3 as it divides 12 four times and 3 is a factor of 12.

But 12 cannot be a multiple of 7 as 12 cannot be divided by 7 completely.

Next we move on to the calculation of the total number of factors.

** ****Total Number of Factors of a Number: **

A composite number N can be written as a product of its prime factors.

If N can be written as then total number of factors will be .

(Where a, b, c are prime numbers)

But where did this formula come from? Let us understand by taking an example.

**Example:** Take N = 300

Prime factors of N=

Now if a number D is to divide N, maximum value of D can be = and minimum can be

Now that we have calculated the total number of factor, let us see how many of those are even and how many are odd.

**Finding ODD and EVEN Factors of a Number: **

**Even Factors:**

**Example:** Let us take an example of N = 240

N= 240=

To make any factor EVEN, 2 or any of its power has to be a multiple of it, otherwise odd*odd can give a odd number. For eg:

2*3=6; *5=20 are even factors, because they are getting multiplied with 2 or with its powers.

But in case we don’t include 2, then

3*5 =15 ; *5 = 5

So, if even factors are to be calculated we need to exclude and include all other powers of 2 while making combinations.

Let D be a possible factor of N, then

D=

In this case, total no of even factors will be = 4 * (1+1)*(1+1) = 16

**Odd factors:**

Again we will take example of N= 240

Unlike the above-discussed solution, in this case only power included of 2 will be

So N= 240=

Odd factors :- 1*(1+1)*(1+1)=4

With everything studied so far, you need to keep the following in mind:

**NOTE 1:**

We know that total no. of factors of 240 will be= 5*2*2=20

Using

In this case =>

Total no of factors = even factors + odd factors

20= 16+4

So after calculating even factors we can subtract them from total number of factors to get number of odd factors.

**NOTE 2:**

Keep in mind that number of odd factors won’t be equal to the number of even factors. Usually we tend to take the number of odd/even factors as half of the total number of factors. But that won’t be the case.

Let’s try some questions based on the above-discussed concepts.

**EXERCISE:**

**Question 1:** Find the total number of factors of 58800.

(1) 90

(2) 80

(3) 180

(4) 45

### Answer and Explanation

**Solution**: option (1)

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

Total number of factors = (4+1)(1+1)(2+1)(2+1)=90

**Question 2:** Find the number of even factors of 58800.

(1) 90

(2) 72

(3) 18

(4) 45

### Answer and Explanation

**Solution**: option (2)

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

To calculate even factors we exclude 2^{0}

So total, number of even factors will be= (4)(1+1)(2+1)(2+1)=72

**Question 3:** Find the number of odd factors of 58800.

(1) 90

(2) 72

(3) 18

(4) 45

### Answer and Explanation

**Solution: option (3)**

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

Number of odd factors = total no of factors – number of even factors

Number of odd factors = 90 -72 = 18

**Question 4:** Find the number of even factors of 840.

(1) 32

(2) 72

(3) 24

(4) 48

### Answer and Explanation

**Solution**: Option (3)

We first factorize 840.

To calculate even factors we exclude 2^{0}

So total, number of even factors will be= (3)(1+1)(1+1)(1+1)=24

**Question 5:** Find the number of odd factors of 840.

(1) 8

(2) 7

(3) 18

(4) 32

### Answer and Explanation

**Solution: Option (1)**

We first factorize 58800.

Total no of factors = 4*2*2*2=32

Number of odd factors = total no of factors – number of even factors

Number of odd factors = 32-24 = 8