SUM OF THE FACTORS OF A NUMBER – MINDSPARK

FACTORS OF A NUMBER

Factors of a number are the numbers that divide the given number completely. Prime numbers only have two factors that are the number itself and 1 whereas composite numbers have more than two factors which also include prime factors. Prime numbers are those numbers that are not divisible by any other number except 1 and the number itself. Non-prime(Composite) numbers are divisible by more than two numbers.

Examples 

13 is a prime number and has only two factors which are 1 and 13. 

36 is a composite number and has more than two factors which are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The ordered pairs are (1 x 36 = 36, 2 x 18 = 36, 3 x 12 = 36, 6 x 6 = 36). Prime factors are the factors that are prime in nature. The number 36 can be expressed as the product of prime factors : 36=3^{2} \times 2^{2}

FORMULAS FOR FACTORS OF A NUMBER

There are basic formulas for factors of a number N, where N =p^{a} q^{b} r^{c} and p, q and r are the prime factors of a number and a, b and c are positive exponents. They are namely – 

1. Number of factors – We can easily find the number of factors of a number N with the help of the following formula: (a+1)(b+1)(c+1)
   
   
2. Sum of the factors – The sum of the factors of a number can be calculated using the formula: \left(\frac{p^{a+1}-1}{p-1}\right)\left(\frac{q^{b+1}-1}{q-1}\right)\left(\frac{r^{c+1}-1}{r-1}\right) 
3. Product of factors – The product of the factors of the number N has the following formula: N^{\text {no. of factors } / 2}

EXAMPLES

Let us look at some examples

1. Consider the number 36.

The prime factorisation of 36 is 36=2^{2} \times 3^{2}, so the number of factors of 36 are ( 2+1)(2+1) = 9

The sum of the factors of 36

=\left(\frac{2^{2+1}-1}{2-1}\right)\left(\frac{3^{2+1}-1}{3-1}\right)

=\frac{7}{1} \times \frac{8}{2}

= 28

The product of the factors is 36^{9 / 2}=1,00,77,696

2. Consider the number 15 = 5 x 3

The number of factors of 15 are (1+1)(1+1) = 4

Sum of the factors is \left(\frac{5^{1+1}-1}{5-1}\right)\left(\frac{3^{1+1}-1}{3-1}\right)=24

Product of the factors is 15^{4 / 2}=225