How To Solve: Number and Sum of Factors of a number

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Following is covered in the video

¤ How to find Number of Factors of a number.
¤ Solved Problems: Number of Factors
¤ How to find Sum of Factors of a number.
¤ Solved Problems: Sum of Factors

How to find Number of Factors of a number.

To find number of factors of a number we need to write the number as product of power of prime number and add one to the powers and multiply the powers.

Example : Find the number of factors of 72.

72 = \(2^3 * 3^2\)
=> Number of factors of 72 = (3+1) * (2+1) = 4 * 3 = 12

We can list down the factors as
1 * 72
2 * 36
3 * 24
4 * 18
6 * 12
8 * 9


Solved Problems: Number of Factors

Q1 : Find number of factors of 100 and list the factors?

Sol: 100 = \(2^2 * 5^2\)
=> Number of factors of 100 = (2+1) * (2+1) = 3 * 3 = 9

We can list down the factors as
1 * 100
2 * 50
4 * 25
5 * 20
10 * 10

Q2 : Find number of factors of 8?

Sol: 8 = \(2^3\)
=> Number of factors of 8 = (3+1) = 4

We can list down the factors as 1, 2, 4, 8


How to find Sum of Factors of a number

Sum of factors of an integer n, where n = \(p^a * q^b * r^c\) (p, q, r being prime number) is given by

\(\frac{(p^{a+1} – 1) * (q^{b+1} – 1) * (r^{c+1} – 1)}{ (p-1) * (q-1) * (r-1)}\)

Solved Problems: Sum of Factors

Q1 : Find the sum of Factors of 60.

Sol: 60 = \(2^2 * 3^1 * 5^1\)
Sum of factors of 60 = \(\frac{(2^{2+1} – 1) * (3^{1+1} – 1) * (5^{1+1} – 1)}{ (2-1) * (3-1) * (5-1)}\) = \(\frac{7 * 8 * 24}{ 1 * 2 * 4}\) = 168

Hope it helps!
Good Luck!