lcm6

lcm6 refers to the least common multiple of six integers. It is the six-argument specialization of the general least common multiple (lcm) function. For integers a1, a2, a3, a4, a5, and a6, lcm6(a1, a2, a3, a4, a5, a6) is the smallest positive integer that is a multiple of each ai.

Computation can be done by prime factorization or by iterative application of the binary lcm operation. If ai is factored as ai = ∏ p_j^{e_{i,j}}, then lcm6 equals ∏ p_j^{max_i e_{i,j}} over all primes p_j appearing in any ai. Alternatively, lcm6 can be computed as lcm(lcm(lcm(lcm(lcm(a1, a2), a3), a4), a5), a6).

Properties of lcm6 follow from the general lcm. The operation is commutative and associative, so the order of arguments does not affect the result. In most computational contexts, if any argument is zero, the value of lcm6 is zero; otherwise the result is positive. Some mathematical conventions treat zero in LCM differently, but the common programming and number theory approach sets lcm(0, x) = 0 for x ≠ 0.

Example: lcm6(8, 12, 9, 5, 7, 14) = 2520, since 2520 is divisible by 8 (2^3), 12 (2^2·3), 9 (3^2), 5, 7, and 14 (2·7). This yields the maximum prime powers 2^3, 3^2, 5, and 7.

Applications include problems requiring a common multiple of six integers, fraction addition with six denominators, and modular arithmetic. See also LCM and the general concept of the least common multiple.