Factors & Prime Factorisation Calculator

Factors and Prime Numbers Guide

What's the key thing to understand about 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.

What's the key thing to understand about 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.

What do I need to know about 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.

What do I need to know about 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.