Sum and Number of Even and Odd Factors of a Number

The number system is an important topic in banking exams. One of the subtopics is finding the sum and number of even and odd factors of a number. This can be done by following the methods explained below.

Method 1: Prime Factorization

- Find the prime factorization of the given number.
- Write all the factors of the number using the prime factors.
- Count the number of even factors (those with at least one factor of 2) and odd factors (those without any factor of 2).
- Add up all the factors to find their sum.

Method 2: Formula

- Use the formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then the number of factors of n is (a1+1)(a2+1)...(ak+1).
- To find the number of even factors, subtract the number of odd factors from the total number of factors.
- Use the formula for sum of factors: If n = p1^a1 * p2^a2 * ... * pk^ak, then the sum of factors of n is [(p1^(a1+1) - 1)/(p1-1)] * [(p2^(a2+1) - 1)/(p2-1)] * ... * [(pk^(ak+1) - 1)/(pk-1)].

Example: Find the sum and number of even and odd factors of 360.

Method 1:

- 360 = 2^3 * 3^2 * 5
- Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
- Even factors: 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 40, 60, 72, 90, 120, 180, 360 (18 even factors).
- Odd factors: 1, 3, 5, 9, 15, 45 (6 odd factors).
- Sum of factors: 1236.

Method 2:

- Number of factors: (3+1)(2+1)(1+1) = 24.
- Number of even factors: 24 - 6 = 18.
- Sum of factors: [(2^4-1)/(2-1)] * [(3^3-1)/(3-1)] * [(5^2-1)/(5-1)] = 1236.

In conclusion, these methods can be used to find the sum and number of even and odd factors of a number in banking exams.