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مرکز یادگیری / حساب و اعداد

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 دقیقه مطالعهبه‌روزرسانی ۲۶ مهر ۱۴۰۵نوشته Sumzo teamبازبینی: 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.

مثال حل‌شده

Prime factors of 360

Write 360 as a product of primes.

نمایش راه‌حل
  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

پاسخ: 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

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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.

مثال حل‌شده

GCD and LCM of 12 and 18

Find the GCD and LCM of 12 and 18.

نمایش راه‌حل
  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

پاسخ: GCD = 6, LCM = 36

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