least common multiple
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: The least common multiple of two (or more) nonzero integers is the least positive integer divisible by all of them. This is usually denoted lcm. For example, lcm(-12,30)=60. Notice Note that we can use the first of these, and the Euclidean algorithm, to find the least common multiple without factoring. However, if we know the prime factorization of a is prod p_sup_i ^ e_sub_i, and that of b is prod p_sup_i ^ f_sub_i, then lcm(a,b) is prod p_sup_i ^ max(3_sub_i,f_sub_i).

We can generalize gcd(a,b).lcm(a,b) = ab to three variables in a number of ways:

  • gcd(a,b,c).lcm(ab,ac,bc) = abc
  • lcm(a,b,c).gcd(ab,ac,bc) = abc
  • gcd(lcm(a,b),lcm(a,c),lcm(b,c)) = lcm(gcd(a,b), gcd(a,c),gcd(b,c))
Finally, as a direct result of the properties of min and max functions we have the dual relations:
  • gcd(a, lcm(b,c)) = lcm(gcd(a,b),gcd(a,c))
  • lcm(a, gcd(b,c)) = gcd(lcm(a,b),lcm(a,c))

See Also: GCD


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