Prime factors, GCD and LCM

Break numbers into primes, then use the prime factorisation to find the greatest common divisor and least common multiple.

1분 분량2026년 10월 18일 업데이트글: Sumzo team검토: Sumzo team

A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, …). Every whole number greater than 1 can be written as a product of primes in exactly one way.

Finding the prime factorisation

Divide by the smallest prime that goes in, and repeat with what is left until you reach 1.

풀이 예제

Prime factors of 360

Write 360 as a product of primes.

풀이 과정 보기

Divide by 2 three times. $360 \to 180 \to 90 \to 45$
Divide by 3 twice, leaving the prime 5. $45 \to 15 \to 5$
Collect the primes. $360 = 2^3 \times 3^2 \times 5$

답: $2^3 \times 3^2 \times 5$ , which has $(3+1)(2+1)(1+1) = 24$ divisors

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GCD and LCM from prime factors

GCD (greatest common divisor, also called HCF): multiply the primes the numbers share, each to its lowest power.
LCM (least common multiple): multiply every prime that appears, each to its highest power.

풀이 예제

GCD and LCM of 12 and 18

Find the GCD and LCM of 12 and 18.

풀이 과정 보기

Factorise both numbers. $12 = 2^2 \times 3,\quad 18 = 2 \times 3^2$
Lowest powers for the GCD, highest powers for the LCM. $\gcd = 2 \times 3 = 6,\quad \operatorname{lcm} = 2^2 \times 3^2 = 36$

답: GCD = 6, LCM = 36