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Mathematics · pattern A, Numeric fields

Lowest common multiple, via the GCF.

Enter two numbers for their lowest common multiple, with the greatest common factor and the Euclidean algorithm steps that produce it.

Inputs

The formula used

LCM(a, b) = |ab| ÷ GCF(a, b)

Computing the GCF first is far quicker than listing multiples, and the relationship between the two is exact.

LCM

36

For 12 and 18: GCF 6, LCM 36.

Prime factors of 12
2 × 2 × 3
Prime factors of 18
2 × 3 × 3
GCF
6
LCM
36
Coprime?
no

Worked steps

  1. 112 = 0 × 18 + 12

    Euclid’s algorithm: replace the pair with (divisor, remainder)

  2. 218 = 1 × 12 + 6

    Euclid’s algorithm: replace the pair with (divisor, remainder)

  3. 312 = 2 × 6 + 0

    Euclid’s algorithm: replace the pair with (divisor, remainder)

  4. 4GCF = 6

    the last non-zero remainder is the greatest common factor

GCF times LCM equals the product of the numbers.

Every prime appears in the GCF at its lowest power and in the LCM at its highest, so the two together account for exactly the primes in both numbers. That is why ab = GCF × LCM always holds, and why finding one immediately gives the other.

Questions about lcm

Where is the LCM used?
Adding fractions with different denominators, and any repeating-cycle problem where two periods coincide.
What is the LCM of coprime numbers?
Their product, because the GCF is 1. For 8 and 15 it is 120.
Does it extend to three numbers?
Yes, pairwise: LCM(a, b, c) = LCM(LCM(a, b), c).

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