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Prime factors, GCD and LCM

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

1 хв читанняОновлено 18 жовтня 2026 р.Автор: Sumzo teamПеревірив(-ла): 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.

Розібраний приклад

Prime factors of 360

Write 360 as a product of primes.

Показати розв’язання
  1. Divide by 2 three times.
    360→180→90→45360 \to 180 \to 90 \to 45
  2. Divide by 3 twice, leaving the prime 5.
    45→15→545 \to 15 \to 5
  3. Collect the primes.
    360=23×32×5360 = 2^3 \times 3^2 \times 5

Відповідь: 23×32×52^3 \times 3^2 \times 5, which has (3+1)(2+1)(1+1)=24(3+1)(2+1)(1+1) = 24 divisors

Перевірено математичним рушієм Sumzo

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.

Розібраний приклад

GCD and LCM of 12 and 18

Find the GCD and LCM of 12 and 18.

Показати розв’язання
  1. Factorise both numbers.
    12=22×3,18=2×3212 = 2^2 \times 3,\quad 18 = 2 \times 3^2
  2. Lowest powers for the GCD, highest powers for the LCM.
    gcd⁡=2×3=6,lcm⁡=22×32=36\gcd = 2 \times 3 = 6,\quad \operatorname{lcm} = 2^2 \times 3^2 = 36

Відповідь: GCD = 6, LCM = 36

Перевірено математичним рушієм Sumzo

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