Number System: Concept of factors I Total number of factors I Even factor I Odd factors I Sum of factors I Product of factors

Whenever we want to know about the factors of a number we look for the divisors. Like 12 has divisors or factors like 1, 2, 3, 4, 6, and 12. So, it is easy to determine the 6 divisors of 12. What if the number is larger? Our difficulty increases and in the examination, we can end up with nothing. So the approach is very important to improve your speed.

We know that we can represent any natural number N as a product of its prime factors as given below:

N = a^p x b^q x c^r

Here a,b,c… are prime factors of N & p,q,r … are the powers or index raised to the prime numbers of N

Therefore, the total number of factors of N is given by-

( p +1)(q+1)(r+1)

Table of Contents

Problem 01:

Find the total number of factors of 360.

Solution:

we can represent, 360 = 2³ x 3² x 5¹

Total number of factors = ( 3 + 1)(2+1)(1+1)

= 4 x 3 x 2 = 24

So, the Factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360.

In this article we are going to learn about the –

Total number of factors
Number of even factors
Number of odd factors
Number of prime factors
Sum of all factors
Product of all factors.

As we can see, 360 has a total of 24 factors, among those factors some are odd and some are even factors.

Let N be any natural number, N = 2^p x b^q x c^r …

Thus, the Total Number of even factors = p ( q +1)(r +1)…. , where p is the power raised to 2 ( which is the only even prime).

Problem 02:

Find the total number of even factors of 360.

Solution:

we can represent, 360 = 2³ x 3² x 5¹

Total number of factors = ( 3 )(2+1)(1+1)

= 3 x 3 x 2 = 18

From the above example, we easily notice the even factors are 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 40, 60, 72, 90, 120, 180 and 360.

The total number of odd factors = ( q +1)(r +1)….

[Note: Not considered the power of 2 as, 2 can convert other odd factors into even]

Problem 03:

Find the total number of odd factors of 360.

Solution:

we can represent, 360 = 2³ x 3² x 5¹

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

= 3 x 2 = 6

So, the odd factors are – 1, 3, 5, 9, 15, and 45.

The total number of prime factors of N is the total number of prime numbers present.

Problem 04:

Find the number of prime factors of 1260.

Solution:

let’s write the number in prime factorisation form, 1260 = 2² x 3² x 5¹x 7¹

So, the total number of prime factors is 4. That is 2, 3, 5, and 7.

Total number of Prime factors in N = p + q + r + …

Problem 05:

Find the total number of prime factors in 1260.

Solution:

As we can write the number in 1260 = 2² x 3² x 5¹x 7¹

So, the total number of prime numbers or factors present in 1260 is ( 2 + 2 +1 +1) = 6.

SUM OF ALL THE FACTORS

If we need to find out the sum of all the factors of a number N = a^p x b^q x c^r x …

Then, Sum ( S ) =

\begin{equation} \frac{a^{p+1}-1}{a-1} \times \frac{b^{q+1}-1}{b-1} \times \frac{c^{r+1}-1}{c-1} \times \ldots \end{equation}

Problem 06:

Find the sum of all the factors of 12.

Solution:

we can represent, 12 = 2² x 3¹

Therefore, SUM (S)

\begin{equation} \begin{aligned} &=\frac{2^{2+1}-1}{2-1} \times \frac{3^{1+1}-1}{3-1} \\ &=\frac{8-1}{2-1} \times \frac{9-1}{3-1} \\ &=\frac{7}{1} \times \frac{8}{2} \\ &=7 \times 4=28 \end{aligned} \end{equation}

We can cross-check that calculation, as the factors of 12 are 1, 2, 3, 4, 6, and 12.

Sum of all the factors = 1 + 2+ 3+ 4 +6 + 12 = 28

So, for large numbers where writing factors are a bit difficult, we can rely on the above-mentioned formula.

SUM OF ALL THE EVEN FACTORS

Working steps are very simple –

Step 1: Divide the number by 2

Step 2: Find the sum of all the factors of the new resultant number

Step 3: Multiply the sum by 2.

Problem 07:

Find the sum of all the even factors of 24.

Solution:

To find the sum of even factors of 24, first, we have to divide the number by 2, we get 12. The next step is to find out the Sum of all the factors of 12 = 22 x 31 we get,

\begin{equation} \begin{aligned} \text { Sum } &=\frac{2^{2+1}-1}{2-1} \times \frac{3^{1+1}-1}{3-1} \\ &=7 \times 4 \\ &=28 \end{aligned} \end{equation}

The sum of all the even factors of 24 is 28 x 2 = 56.

Let’s cross-check,

Even factors of 24 are 2, 4, 6, 8, 12, and 24.

The Sum of all the even factors of 24 is ( 2 + 4 +6 +8 + 12 + 24) = 56.

So, the working step will help you to determine the even factors of any large number.

PRODUCT OF ALL THE FACTORS

Product of all the factors given by – N^n(F)/2

Working Steps:-

Write down the Number ( N ) into N = a^p x b^q x c^rx ..
Find the total number of factors n(F)
The product of all factors by the formula N^n(F)/2

Problem 08:

Find the product of all the factors of 24

Solution:

24 = 2³ x 3¹

Total number of factors = ( 3+1) ( 1+1) = 4 x 2 = 8

Then,the product of all the factors = 24^8/2= 24⁴ = 331776

Let’s cross-check the formula.

Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

= 1 x 2 x 3 x 4 x 6 x 8 x 12 x 24

= ( 1x 24) ( 2 x12) ( 3 x 8)(4 x 6)

= 24 x 24 x 24 x 24

= 24⁴ = 331776

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Number System: Concept of factors I Total number of factors I Even factor I Odd factors I Sum of factors I Product of factors