Factor Calculator

Every whole number can be broken down into the smaller numbers that divide it evenly — its factors. This calculator lists all of them, pairs them up, counts them, sums them, and tells you whether the number is prime, composite, or even a perfect number.

Students use it across number theory and fraction work; puzzle solvers and programmers use it when divisibility matters. The factor pairs are especially handy for problems like finding rectangle dimensions for a given area, and the perfect-number check adds a touch of mathematical curiosity that pure factor lists miss.

Factors are found by testing every integer from one up to the square root of the target. Each divisor discovered yields two factors at once — the divisor itself and the quotient — so checking beyond the square root would only rediscover pairs already found in reverse.

That optimisation is what makes the search practical. Finding the factors of a million requires a thousand tests rather than a million, and the saving grows with the size of the number. Perfect squares are the one case where a divisor pairs with itself, so the tool avoids listing it twice.

Factors are presented in pairs as well as in a flat sorted list, since the pairing shows the structure: every factor below the square root has exactly one partner above it, and their product is always the original number.

How many factors a number has follows from its prime decomposition: add one to each prime's exponent and multiply those together. So 24, being 2³ × 3, has (3+1)(1+1) = 8 factors. The rule also explains why only perfect squares have an odd factor count — every exponent must be even for the product of the incremented exponents to be odd.

Numbers with unusually many factors relative to their size are called highly composite, and 12, 24, 60, and 360 are among them. That abundance of divisors is exactly why those numbers were chosen for hours, minutes, degrees, and old currency systems: they divide evenly in many more ways than ten does.

The difficulty of the reverse problem is what modern cryptography rests on. Multiplying two large primes is trivial; recovering them from the product is not, and no efficient classical algorithm is known. Trial division of the kind used here is fine for everyday numbers and hopeless beyond a certain size — which is the whole basis of RSA encryption.