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GCD and LCM

Euclid's algorithm and the identity gcd × lcm = a × b.

The greatest common divisor is found without factoring at all: replace the larger number by the remainder when it is divided by the smaller, repeat, and the last non-zero remainder is the gcd. That is Euclid's algorithm, 2,300 years old and still what your computer uses. The lcm follows from gcd × lcm = a × b.

Worked example · gcd(48, 18)

Number-Theory · Gcd-And-Lcm

Gcd(48, 18)

48,\ 18

Step by step

  1. \gcd(48, 18)

    Use Euclid's algorithm: replace the larger number by its remainder on division by the smaller, until the remainder is 0.

  2. 48 = 2 \times 18 + 12

    Divide 48 by 18: quotient 2, remainder 12.

  3. 18 = 1 \times 12 + 6

    Divide 18 by 12: quotient 1, remainder 6.

  4. 12 = 2 \times 6 + 0

    Divide 12 by 6: quotient 2, remainder 0.

  5. \gcd(48, 18) = 6

    The last non-zero remainder is 6.

Reveal the answer
\gcd(48, 18) = 6

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How to: GCD and LCM

  1. Use Euclid's algorithm: replace the larger number by its remainder on division by the smaller, until the remainder is 0.
  2. Divide 48 by 18: quotient 2, remainder 12.
  3. Divide 18 by 12: quotient 1, remainder 6.
  4. Divide 12 by 6: quotient 2, remainder 0.
  5. The last non-zero remainder is 6.

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller-Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

More in Number Theory

Prime factorisationPrime numbersModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory