In the first part of this article, we have covered the general formula for the number of factors of a number. In this one, we are going to cover how to calculate the number of even and odd factors of a number. You can visit the Part 1 of the series here – https://blog.gmatwhiz.com/the-logic-behind-number-of-factors-i/

Number of odd factors

Now that we understand how we find the number of factors, let us understand how can we find out how many of them are odd and how many of them are even. Let us bring back the example of 24 to do this. Again 24 can be prime factorized as below –

24 = 2^{3} x 3^{1}

Out of all the factors (1, 2, 3, 4, 6, 8, 12, and 24), only 1 and 3 are odd, and the rest are even.

If you look closely, the similarity between factors 1 and 3 is that to obtain them we have used the power 2^{0}, the only odd number present along the top row. We have then multiplied it with powers of 3 and they are always odd so we end up getting an odd number.

It is quite evident that for any other row we are getting even factors. So, to an odd factor, we must have powers of 2 as ‘0’ and ignore the other powers.

So, do we need to do all of these to find out the number of odd factors? No.

We can generalize the formula for the total number of odd factors as below,

If a number N can be prime factorized as

N = (2)^{a} X (P_{1})^{b} X (P_{2})^{c} ……

(Where P_{1}, P_{2}, P_{3} and so on are distinct prime numbers apart from 2) then the

number of odd factors = (b+1) x (c+1) ……

The idea is to ignore the power of prime number 2 and treat the remaining number as a new number and find out its factors of it.

Let’s take an example to understand the process.

What is the number of odd factors of 120?

120 can be prime factorized as 120 = 2^{3} x 3^{1} x 5^{1}

Now we will completely ignore the power of 2.

Thus, the number will turn to 3^{1} x 5^{1}

Now, the number of factors of this number = (1 + 1) x (1 + 1) = 2 x 2 = 9

Number of even factors

Although there is a formula for this, we should not be worried about it at all.

The number of factors can be either odd or even.

By now we know how to find the total number of factors and the number of odd factors. So, finding the number of even factors is no rocket science.

Number of even factors = Total number of factors – Number of odd factors

Let’s take an example:

Question: what is the number of even factors of 72?

Solution:

72 can be prime factorized as 72 = 2^{3} x 3^{2}

Total number of factors = (3 + 1) x (2 + 1) = 4 x 3 = 12

Number of odd factors = (2 + 1) = 3

Thus, number of even factors = 12 – 3 = 9

Before we finish off this article, let’s take one last example of 2700 and find its total number of factors, number of odd factors, and number of even factors.

Table of Contents

In the first part of this article, we have covered the general formula for the number of factors of a number. In this one, we are going to cover how to calculate the number of even and odd factors of a number. You can visit the Part 1 of the series here – https://blog.gmatwhiz.com/the-logic-behind-number-of-factors-i/

Number of odd factorsNow that we understand how we find the number of factors, let us understand how can we find out how many of them are odd and how many of them are even. Let us bring back the example of 24 to do this. Again 24 can be prime factorized as below –

24 = 2^{3}x 3^{1}Out of all the factors (1, 2, 3, 4, 6, 8, 12, and 24), only 1 and 3 are odd, and the rest are even.

^{0}, the only odd number present along the top row. We have then multiplied it with powers of 3 and they are always odd so we end up getting an odd number.So, do we need to do all of these to find out the number of odd factors? No.

We can generalize the formula for the total number of odd factors as below,

If a number N can be prime factorized as

N = (2)^{a}X (P_{1})^{b}X (P_{2})^{c}……(Where P

_{1}, P_{2}, P_{3}and so on are distinct prime numbers apart from 2) then thenumber of odd factors = (b+1) x (c+1) ……Let’s take an example to understand the process.

What is the number of odd factors of 120?

120 can be prime factorized as 120 = 2

^{3}x 3^{1}x 5^{1}^{1}x 5^{1}Number of even factorsAlthough there is a formula for this, we should not be worried about it at all.

By now we know how to find the total number of factors and the number of odd factors. So, finding the number of even factors is no rocket science.

Number of even factors = Total number of factors – Number of odd factorsLet’s take an example:

Question: what is the number of even factors of 72?

Solution:

72 can be prime factorized as 72 = 2

^{3}x 3^{2}Thus, number of even factors = 12 – 3 = 9

Before we finish off this article, let’s take one last example of 2700 and find its total number of factors, number of odd factors, and number of even factors.

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