🧩 Factors Calculators

What Are Factors?

A factor of a positive integer n is any integer that divides n exactly with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 — each divides 12 evenly. Every integer greater than 1 has at least two factors: 1 and itself. Numbers with exactly two factors (1 and themselves) are prime numbers. Numbers with more than two factors are composite. The number 1 is neither prime nor composite — it has only one factor. To find all factors efficiently, only test divisors up to √n; for every factor d ≤ √n, n/d is a corresponding factor ≥ √n. The total number of factors of n can be calculated from its prime factorisation: if n = p₁^a₁ × p₂^a₂ × ..., then the factor count is (a₁+1)(a₂+1)...

Factor Pairs and Their Applications

A factor pair of n is a pair of positive integers (a, b) such that a × b = n. For n = 36, the factor pairs are (1,36), (2,18), (3,12), (4,9), (6,6). Factor pairs are used in area problems — finding all rectangles with integer dimensions and a given area — and in cryptography, where the difficulty of factoring large numbers (finding factor pairs) underlies RSA encryption. A perfect square has an odd number of factors because one factor pair has both elements equal (6×6 = 36), so it is counted once rather than twice. Factor pairs are also used in mental arithmetic: to multiply 36 by 25, recognise that 25 = 100/4, so 36 × 25 = 3600/4 = 900.

Divisibility Rules

Divisibility rules allow quick mental checks without long division. A number is divisible by 2 if its last digit is even; by 3 if the sum of its digits is divisible by 3; by 4 if its last two digits form a number divisible by 4; by 5 if it ends in 0 or 5; by 6 if divisible by both 2 and 3; by 8 if its last three digits are divisible by 8; by 9 if the digit sum is divisible by 9; by 10 if it ends in 0. The rule for 7 is less elegant: double the last digit, subtract from the remaining digits, and repeat — if the result is divisible by 7, so is the original number. These rules are consequences of the decimal positional system and modular arithmetic.

Perfect, Abundant, and Deficient Numbers

A perfect number equals the sum of its proper divisors (all factors except itself). The first perfect number is 6 = 1 + 2 + 3; the next is 28 = 1 + 2 + 4 + 7 + 14. All known perfect numbers are even, and each corresponds to a Mersenne prime: n = 2^(p-1) × (2^p − 1) when (2^p − 1) is prime. Whether any odd perfect numbers exist is one of the oldest unsolved problems in mathematics. An abundant number has a proper divisor sum exceeding the number itself (e.g., 12: 1+2+3+4+6=16>12). A deficient number has a proper divisor sum less than itself (e.g., 8: 1+2+4=7<8). Most integers are deficient; the density of abundant numbers is about 24.76% of all positive integers.