Sum of Factors: Formula, Methods and Solved Examples
Sum of Factors is the total of all positive divisors of a number. It can be found by listing the factors for small numbers or by using prime factorisation for larger numbers. The standard sum of divisors formula uses the powers of the prime factors and converts the calculation into a product of geometric sums.
What Is the Sum of Factors?
If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, where the p values are distinct primes, then the sum of positive divisors is σ(n) = (1 + p₁ + p₁² + ... + p₁ᵃ¹)(1 + p₂ + p₂² + ... + p₂ᵃ²) ... (1 + pₖ + pₖ² + ... + pₖᵃᵏ). For example, 12 = 2² × 3, so σ(12) = (1 + 2 + 4)(1 + 3) = 21. Its positive divisors are 1, 2, 3, 4, 6 and 12, whose sum is also 21.
Sum of Factors Formula & Tricks
Important Formulas
For each distinct prime factor pᵢ, add its powers from pᵢ⁰ to pᵢᵃⁱ, then multiply the resulting sums.
This is the direct form of the formula and is usually convenient for numerical calculations.
A prime number has only two positive divisors: 1 and the number itself.
For a number having only one distinct prime factor, the sum contains all powers of that prime from exponent 0 to a.
Quick Tricks
Factorise the number, add the powers of each prime factor, and multiply the sums. This avoids missing or repeating divisors.
For a prime p with exponent a, 1 + p + p² + ... + pᵃ = (pᵃ⁺¹ − 1)/(p − 1).
Positive divisors can be paired as d and n/d. For a non-square number, every pair is different; for a perfect square, the square root appears only once.
Sum of Factors Concepts
Prime Factorisation Method
If n = pᵃ × qᵇ, then σ(n) = (1 + p + p² + ... + pᵃ)(1 + q + q² + ... + qᵇ). Each term in the product represents one possible divisor formed by choosing a power of each prime.
Sum of Factors of a Prime Power
A prime power has divisors consisting only of the powers p⁰, p¹, p², ..., pᵃ. The geometric-series form is σ(pᵃ) = (pᵃ⁺¹ − 1)/(p − 1).
Multiplicative Property for Coprime Numbers
This property applies when m and n have no common prime factor. It follows because every positive divisor of mn can be formed by multiplying a divisor of m by a divisor of n.
Listing Factors and Pairing Method
Every divisor d of n has a paired divisor n/d. If n is not a perfect square, the pairs contain two different numbers. If n is a perfect square, the square root is the unpaired middle divisor and must not be counted twice.
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Sum of Factors in Number System
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Sum of Factors: Quick Revision
Use these rules to calculate the sum of positive divisors accurately.
- For n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, σ(n) = ∏[(pᵢᵃⁱ⁺¹ − 1)/(pᵢ − 1)].
- The expanded form is the product of sums of powers of each distinct prime factor.
- A prime p has factor sum 1 + p.
- For a prime power pᵃ, the factor sum is 1 + p + ... + pᵃ.
- If gcd(m,n) = 1, then σ(mn) = σ(m)σ(n).
- When listing factor pairs of a perfect square, count the square root only once.
- The standard formula includes positive divisors, including 1 and n.
Sum of Factors FAQs
What is the sum of factors formula for n = pᵃqᵇ?
The formula is σ(n) = (1 + p + p² + ... + pᵃ)(1 + q + q² + ... + qᵇ), or σ(n) = [(pᵃ⁺¹ − 1)/(p − 1)] [(qᵇ⁺¹ − 1)/(q − 1)].
What is the sum of factors of 24?
24 = 2³ × 3, so σ(24) = (1 + 2 + 4 + 8)(1 + 3) = 15 × 4 = 60.
What is the sum of factors of 100?
100 = 2² × 5². Therefore, σ(100) = (1 + 2 + 4)(1 + 5 + 25) = 7 × 31 = 217.
How is the sum of factors of a prime number calculated?
A prime p has only the positive divisors 1 and p. Hence, its sum of factors is 1 + p; for example, σ(13) = 14.
What is the difference between the sum of factors and the sum of proper factors?
The sum of factors includes 1 and n, while the sum of proper factors excludes n. Thus, the sum of proper factors of n is σ(n) − n.
What is the sum of factors of 1?
The only positive divisor of 1 is 1, so σ(1) = 1. The prime-factorisation product is treated as an empty product in this special case.