Factors & Prime Factorisation Calculator
Find every factor of any number, its prime factorisation, and all factor pairs. Shows full working for GCSE and A-level maths.
Factors and Prime Numbers Guide
Finding All Factors
A factor of n is any integer that divides n exactly (with no remainder). To find all factors: test every integer from 1 to √n. If i divides n, both i and n/i are factors. Example for 36: √36 = 6. Test 1 (36/1=36✓), 2 (36/2=18✓), 3 (36/3=12✓), 4 (36/4=9✓), 5 (no), 6 (36/6=6✓). Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. Note: 6 appears only once because 36/6=6.
Prime Factorisation
Every integer greater than 1 can be written as a unique product of prime numbers — this is the Fundamental Theorem of Arithmetic. To find prime factorisation, divide by the smallest prime (2) repeatedly, then try 3, 5, 7, etc. Example: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2 × 2 × 2 × 3 × 15 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5. Uses: simplifying fractions, finding HCF and LCM, solving equations in number theory.
Highly Composite Numbers
Some numbers have unusually many factors because of their prime factorisation structure. 12 = 2² × 3 has 6 factors. 24 = 2³ × 3 has 8. 60 = 2² × 3 × 5 has 12. 360 = 2³ × 3² × 5 has 24 factors — this is why 360 was chosen for degrees in a circle: it is divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. Similarly 12, 24, and 60 for hours, months, and minutes.
The Number of Factors Formula
If n = p₁^a × p₂^b × p₃^c..., the number of factors = (a+1)(b+1)(c+1)... For 360 = 2³ × 3² × 5¹: (3+1)(2+1)(1+1) = 4×3×2 = 24 factors. This formula explains why square numbers always have an odd number of factors (because one factor pair has equal values), while all other numbers have an even count. A prime number always has exactly 2 factors: 1 and itself.
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