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.
Prime Factorization Guide
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...
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: simp
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
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
Recommended for this calculator