The rule
n = p₁^a × p₂^b × … · number of divisors = (a+1)(b+1)…
Why it works
Every whole number greater than one is a product of primes, in exactly one way. That uniqueness is the fundamental theorem of arithmetic, and it is what makes the factorisation a genuine fingerprint of the number rather than one description among many.
How to do it by hand
- Divide by 2 as many times as it goes
- Move to 3, then 5, then each prime in turn
- Stop when the quotient reaches 1
- Count the divisors by adding one to each exponent and multiplying
What is worth knowing
The divisor count is a neat consequence: 360 = 2³ × 3² × 5 gives (3+1)(2+1)(1+1) = 24 divisors, because each divisor picks an exponent for each prime independently. The uniqueness of the factorisation is also why 1 is excluded from the primes — allowing it would let you write 6 as 2 × 3, or 1 × 2 × 3, or 1 × 1 × 2 × 3, and the theorem would collapse. Factoring large numbers is genuinely hard, and that difficulty is what RSA encryption rests on: multiplying two 300-digit primes takes microseconds, and undoing it takes longer than the age of the universe.