Number System part 1 : Factors

Factors of a number is an important subtopic from number systems.

 In this article, we will discuss the fundamentals of factors of a number. Almost every competitive examination has 2-3 medium to difficult level question based on factors of a number. Taking this in consideration, we will discuss in detail the advanced application of this topic, to give you an edge over other candidates.

What are Factors of a number?

Factors of a number N refers to all the numbers which divide N completely. These are also called divisors of a number.

Basic formula related to factors of a number

These are certain basic formulas pertaining to factors of a number N, such that,

N= paqbrc

Where, p, q and r are prime factors of the number n.

a, b and c are non negative powers/ exponents

• Number of factors of N = (a+1)(b+1)(c+1)
• Product of factors of N = N No. of factors/2
• Sum of factors: ( p0+p1+...+pa) ( q0+ q1+....+qb) (r0+r1+...+rc)/ (pa-1)(qb-1)(rc-1)

Solved questions on Factors of a number

Example 1: Consider the number 120. Find the following for n

1. Sum of factors
2. Number of factors
3. Product of factors

Solution: The prime factorization of 120 is 23x31x51. By applying the formulae,

• Sum of factors = [(20+21+22+23)(30+31)(50+51)]/ [(2-1) (3-1)(5-1)] = 45
• Number of factors = (3+1)(1+1)(1+1) = 16
• Product of factors = 120(16/2) = 1208

Example 2: Find the following for the number 84 :-

1. Number of odd factors
2. Number of even factors

Solution: By the prime factorization of 84, 84= 22 × 31 × 71

Total number of factors = (2+1)(1+1)(1+1) = 12

1. Number of odd factors will be all possible combinations of powers of 3 and 5 (excluding any power of 2) . Hence number of odd factors = (1+1)(1+1) = 4
   By manually checking, these factors are 1, 3, 7 and 21.
2. Number of even factors = total no. of factors - no. of even factors
   = 12 - 4 = 8