Prime Factorization, HCF & LCM Calculator
Find prime factors of any number and calculate HCF (Highest Common Factor) and LCM (Lowest Common Multiple) with full factor tree working.
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Prime Factorization Guide
What should I know about Prime Factorization?
Every integer greater than 1 is either prime or can be written as a unique product of primes (Fundamental Theorem of Arithmetic). Method: divide by smallest prime (2) repeatedly, then 3, then 5, etc. 360 ÷ 2 = 180. 180 ÷ 2 = 90. 90 ÷ 2 = 45. 45 ÷ 3 = 15. 15 ÷ 3 = 5. 5 is prime. 360 = 2³ × 3² × 5. Index notation: use powers. 2³ × 3² × 5¹ = 8 × 9 × 5 = 360. Prime numbers have no factors other than 1 and themselves: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
What do I need to know about HCF — Highest Common Factor?
HCF = the largest number that divides exactly into all given numbers. Method using prime factors: list prime factors of each number. HCF = product of COMMON prime factors (lowest power). 360 = 2³ × 3² × 5. 84 = 2² × 3 × 7. Common factors: 2² (min of 3 and 2) × 3¹ (min of 2 and 1) = 4 × 3 = 12. HCF(360, 84) = 12. Alternative: Euclidean algorithm. Divide larger by smaller, take remainder, repeat. 360 = 4×84 + 24. 84 = 3×24 + 12. 24 = 2×12 + 0. HCF = 12 (last non-zero remainder). Applications: simplifying fractions to their lowest terms by dividing both numerator and denominator by their HCF.
What should I know about LCM — Lowest Common Multiple?
LCM = smallest number divisible by all given numbers. Method using prime factors: list prime factors. LCM = product of ALL prime factors at their HIGHEST power. 360 = 2³ × 3² × 5. 84 = 2² × 3 × 7. LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2,520. Key relationship: HCF × LCM = product of the two numbers. 12 × 2520 = 30,240 = 360 × 84 ✓. Applications: finding the lowest common denominator when adding fractions. Scheduling problems (two events that repeat every n and m days — when next coincide?). In music theory, LCM is used to work out when rhythmic patterns of different lengths will align back to their starting point together.
What do I need to know about Applications of HCF and LCM?
HCF applications: tiling — what is the largest square tile that fits exactly in a 360cm × 84cm room? HCF(360,84) = 12cm tiles. Simplifying fractions: 84/360 = (84÷12)/(360÷12) = 7/30. Sharing: split 360 items and 84 items into equal groups as large as possible = 12 groups of 30 and 7. LCM applications: when will two buses (one every 15 mins, one every 35 mins) next leave together? LCM(15,35) = 105 minutes. Adding fractions: 1/9 + 1/6 = LCM(9,6) = 18. Denominator = 18. 2/18 + 3/18 = 5/18. Cyclic events — anything that repeats on a fixed schedule, from bus timetables to planetary alignments — rely on LCM to determine when multiple cycles will next coincide.