Factor Calculator

Enter any positive integer to find all its factors in pairs, its prime factorization, the total number of factors, and whether it is perfect, abundant, or deficient.

Enter a positive integer (up to 1,000,000).

Number

What is a Factor?

A factor of a whole number is any integer that divides into that number exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 — because each of these numbers divides 12 without a remainder. Factors always come in pairs: if 3 is a factor of 12, then 12 ÷ 3 = 4 means 4 is also a factor. These are called factor pairs. The total number of factors a number has is related to its prime factorisation — a number with many small prime factors tends to have many total factors.

Prime factorisation expresses a number as a product of prime numbers raised to their respective powers. By the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorisation (up to the order of factors). For example, 360 = 2³ × 3² × 5. The total number of factors can be computed directly from the prime factorisation: add 1 to each exponent and multiply the results — for 360, this gives (3+1)(2+1)(1+1) = 24 factors. This shortcut is far faster than listing factors individually for large numbers.

Factorisation underpins many areas of mathematics and computing. In cryptography, the difficulty of factoring very large semiprimes (products of two large primes) is the security foundation of RSA encryption. In simplifying fractions, finding common factors of the numerator and denominator allows reduction to lowest terms. In number theory, a number's factorisation reveals its classification: perfect numbers (like 6 and 28) have proper divisor sums equal to themselves; abundant numbers have sums that exceed them; deficient numbers have sums below them. This calculator finds all factors, computes the prime factorisation, and classifies the number for complete factorisation analysis.

Classifications Explained

Perfect:Sum of proper divisors equals the number (e.g. 6 = 1+2+3).

Abundant:Sum of proper divisors exceeds the number (e.g. 12: 1+2+3+4+6 = 16 > 12).

Deficient:Sum of proper divisors is less than the number (e.g. 8: 1+2+4 = 7 < 8).

How the Factor Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is a factor?

A factor (or divisor) of an integer n is any integer that divides n without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

What is prime factorization?

Prime factorization expresses a number as a product of prime numbers. Every integer greater than 1 has a unique prime factorization. For example, 360 = 2³ × 3² × 5.

How many factors does a number have?

If n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ, then the number of factors is (a₁+1)(a₂+1)…(aₖ+1). For 360 = 2³ × 3² × 5¹, that is (3+1)(2+1)(1+1) = 24 factors.

What is a perfect number?

A perfect number equals the sum of its proper divisors. The first few perfect numbers are 6, 28, 496, and 8128. Only 51 perfect numbers are known, and all known ones are even.

What is the difference between a factor and a multiple?

A factor divides evenly into a number. A multiple is the result of multiplying a number by an integer. For 4: factors are 1, 2, 4; multiples are 4, 8, 12, 16…

Real-World Applications

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Cryptography (RSA)

RSA encryption relies on the fact that factoring very large numbers (1,024-bit semiprimes) is computationally infeasible — factorisation is the security foundation.

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Simplifying Fractions

Finding the GCD of numerator and denominator through prime factorisation allows fractions to be reduced to their lowest terms.

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Tile & Grid Planning

Finding factors of a room's dimensions determines which square tile sizes will fit without cutting — factors are the valid sizes.

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Scheduling & Timetabling

LCM (the inverse application of factorisation) finds when two repeating events coincide — e.g. two buses with 12- and 18-minute cycles meet every 36 minutes (LCM).

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Number Theory Research

Prime factorisation determines whether a number is perfect, abundant, or deficient — classifications with deep roots in mathematical history.

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Algorithm Analysis

The Fundamental Theorem of Arithmetic underpins algorithms in number theory, hashing, and checksum generation in computer science.

Common Mistakes

Treating 1 as a prime number

1 is NOT prime — prime numbers must have exactly two distinct positive divisors (1 and itself). 1 has only one divisor, so by definition it is neither prime nor composite.

Stopping prime factorisation too early

When factorising 12 = 4 × 3, students sometimes stop rather than continuing to fully prime factorised form: 2² × 3. Every composite factor must be broken down further.

Confusing factors with multiples

Factors divide INTO a number; multiples are produced BY multiplying a number. Factors of 12: {1,2,3,4,6,12}. Multiples of 12: {12,24,36,…}.

Missing factor pairs by stopping at the square root

Systematic factor-pair listing should stop at √n — all factor pairs above √n are mirrors of those below. Missing this leads to an incomplete factor list.

Assuming large numbers have no small prime factors

Always test divisibility by 2, 3, 5, and 7 first — many large numbers are divisible by these small primes, making full factorisation much simpler.

Number Classification by Factor Sum

Classification	Condition	Example
Perfect	Sum of proper divisors = n	6 (1+2+3=6), 28 (1+2+4+7+14=28)
Abundant	Sum of proper divisors > n	12 (1+2+3+4+6=16 > 12)
Deficient	Sum of proper divisors < n	9 (1+3=4 < 9)
Prime	Only divisors are 1 and itself	7, 13, 17, 19, 23
Composite	Has at least one factor besides 1 and itself	4, 6, 8, 9, 10
1 (special case)	Neither prime nor composite	Divisors: {1} only

References

Hardy, G.H. and Wright, E.M. An Introduction to the Theory of Numbers. Oxford University Press, 2008.
Niven, Ivan et al. An Introduction to the Theory of Numbers. Wiley, 1991.
Rivest, R., Shamir, A., Adleman, L. "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems." Communications of the ACM, 1978.
Knuth, Donald. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. Addison-Wesley, 1997.
National Institute of Standards and Technology. Digital Signature Standard (FIPS 186-5). NIST, 2023.