Number of Factors: Formula, Method and Examples
Number of Factors means the count of positive divisors of a number. The count is found by writing the number as a product of prime powers and multiplying one more than each exponent. This page covers the divisor count formula, factor-counting method, special cases, shortcuts and numerical examples.
What Is the Number of Factors?
The exponents are obtained from the prime factorisation of n. For example, 360 = 2³ × 3² × 5¹, so the number of factors is (3 + 1)(2 + 1)(1 + 1) = 24. Each divisor is formed by independently choosing an exponent from 0 to the exponent available for every prime.
Number of Factors Formula & Tricks
Important Formulas
For each prime pᵢ, its exponent in a divisor can have aᵢ + 1 possible values: 0, 1, 2, ..., aᵢ. Multiplying these choices gives the total number of factors.
A prime number has exactly two positive factors: 1 and the prime number itself.
The factors of pᵃ are 1, p, p², ..., pᵃ, giving a + 1 factors.
Factors usually occur in pairs. A perfect square has one unpaired factor, its square root, so its divisor count is odd.
Quick Tricks
Do not list all factors. First write the number in prime-factorised form, then add 1 to each exponent and multiply the results.
A number has an odd number of positive factors exactly when it is a perfect square. This follows because only the square root pairs with itself.
For a non-square number, factors occur in pairs and the total count is twice the number of factor pairs. For a square number, the square-root pair is counted only once.
Number of Factors Concepts
Prime Factorisation Method
If n = pᵃ × qᵇ × rᶜ, then d(n) = (a + 1)(b + 1)(c + 1). The primes must be distinct, and their exponents must be read from the complete prime factorisation.
Factors of a Prime Power
The possible factors are p⁰, p¹, p², ..., pᵃ. For example, 2⁵ has the factors 1, 2, 4, 8, 16 and 32, so it has 6 factors.
Odd and Even Number of Factors
For a non-square n, each factor x pairs with a different factor n/x. For n = m², the factor m pairs with itself, leaving one unpaired factor and making the total odd.
Factors and Proper Factors
If d(n) is the total number of positive factors of n, then the number of proper positive factors is d(n) − 1 for n > 1. For n = 1, the only positive factor is 1, and it has no proper positive factor.
Numbers with a Fixed Number of Factors
For example, a number with exactly 6 factors can have the exponent patterns 5 or 2 × 1. Therefore, it can be of the form p⁵ or p²q, where p and q are distinct primes.
Number of Factors Video Lessons
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Multiples and Factors in Number System
Understand the meaning of multiples and factors, learn how they are related, and identify them using simple examples in the Number System.
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Number of Factors: Quick Revision
Use these rules to calculate divisor counts quickly and avoid common counting errors.
- If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then d(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1).
- A prime number has exactly 2 positive factors.
- The number 1 has exactly 1 positive factor.
- A prime power pᵃ has a + 1 positive factors.
- A number has an odd number of factors if and only if it is a perfect square.
- The number of proper positive factors of n > 1 is d(n) − 1.
- For a non-square number, factors occur in distinct pairs; for a square, the square-root factor is unpaired.
Number of Factors FAQs
What is the number of factors formula?
If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then the number of positive factors is d(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1).
How many factors does 360 have?
360 = 2³ × 3² × 5. Therefore, d(360) = (3 + 1)(2 + 1)(1 + 1) = 24.
How many factors does a prime number have?
Every prime number has exactly 2 positive factors: 1 and the number itself.
Why does a perfect square have an odd number of factors?
Factors occur in pairs as x and n/x. For n = m², the factor m pairs with itself, so one factor is unpaired and the total count is odd.
How many factors does 1 have?
The number 1 has exactly 1 positive factor, which is 1 itself.
How many proper factors does 48 have?
Since 48 = 2⁴ × 3, it has d(48) = (4 + 1)(1 + 1) = 10 positive factors. Excluding 48 itself leaves 9 proper positive factors.