Prime Factorisation Calculator

One number, one factorisation

360 = 2³ × 3² × 5, and there is no other way to write it as a product of primes. That uniqueness is the fundamental theorem of arithmetic, and it is the reason 1 is deliberately excluded from the primes — if 1 counted, you could pad any factorisation with as many 1s as you liked and uniqueness would collapse.

Counting divisors without listing them

Take the exponents in the factorisation, add one to each, multiply them together. For 360 = 2³ × 3² × 5¹ that is 4 × 3 × 2 = 24 divisors. The logic is simple: each divisor picks an independent power of each prime, from zero up to the exponent. This is far faster than trial division and is how the calculator gets the count on large numbers.

Why you only test up to the square root

If a number has a factor larger than its square root, the matching co-factor must be smaller than it — and you would already have found that one. So checking divisors up to √n is enough to settle primality. For 1,000,003 that is around a thousand checks instead of a million, and the saving grows enormously as numbers get larger.

Where this stops being a toy

Multiplying two large primes is fast; recovering them from the product is not. That asymmetry is what RSA encryption rests on. The numbers used there run to 600 digits or more — far past anything a browser will factor — and the practical impossibility of reversing the multiplication is precisely the security guarantee.

Frequently Asked Questions