Factors and Prime Factorisation
A factor is any whole number that divides evenly into another with no remainder. Every number above 1 has at least two — itself and one — and a number with exactly those two is prime.
The fundamental theorem of arithmetic states that every integer above 1 has exactly one prime factorisation. 360 is 2³ × 3² × 5, and no other combination of primes produces it. This uniqueness is what makes prime factorisation such a useful tool.
Counting Factors Without Listing Them
Take the prime factorisation, add one to each exponent, and multiply:
360 = 2³ × 3² × 5¹ → (3+1)(2+1)(1+1) = 24 factors
The logic is straightforward: each factor is built by choosing how many of each prime to include — 0 to 3 twos, 0 to 2 threes, 0 or 1 five.
Factor Pairs
Factors come in pairs multiplying to the original number. For 360: 1×360, 2×180, 3×120, 4×90, 5×72, 6×60, 8×45, 9×40, 10×36, 12×30, 15×24, 18×20. This is why testing for factors only requires going up to the square root — every factor above it is paired with one below.
Special Kinds of Numbers
| Type | Definition | Examples |
|---|---|---|
| Prime | Exactly two factors | 2, 3, 5, 7, 11, 13 |
| Composite | More than two factors | 4, 6, 8, 9, 10 |
| Perfect square | An odd number of factors | 1, 4, 9, 16, 25 |
| Perfect number | Proper factors sum to itself | 6, 28, 496, 8128 |
| Highly composite | More factors than any smaller number | 12, 24, 36, 48, 60 |
One is neither prime nor composite — a definition chosen precisely so that prime factorisation stays unique. Perfect squares are the only numbers with an odd factor count, because one pair is a number multiplied by itself.
Why Factorisation Is Hard
Multiplying two large primes is instant; recovering them from the product is not. RSA encryption rests entirely on that asymmetry. Factoring a 2048-bit number is beyond current computers, which is why RSA keys of that size remain in use — and why Shor's algorithm on a sufficiently large quantum computer would break them.
Frequently Asked Questions
Is 1 a prime number?
No, by definition. Including it would break the uniqueness of prime factorisation, since you could insert any number of ones.
What is the fastest way to find factors by hand?
Test divisors from 1 up to the square root, recording both members of each pair. For 360 you only need to test up to 18.
Do factors have to be positive?
Conventionally yes in school mathematics. In number theory, negative divisors exist too — −3 divides 12 — but the convention is to list positive factors only.