 Least common multiple

For positive integers $a,b$, their least common multiple is the smallest positive integer that is an integer multiple of both. Write it as $\mathrm{lcm}(a,b)$. A positive common integer multiple always exists, since $ab$ is one, but it can be larger than necessary.

For example, the positive integer multiples of $12$ begin $12,24,36,48,60,72$, while those of $18$ begin $18,36,54,72$. Their first positive coincidence is $36$, so $\mathrm{lcm}(12,18)=36$. Both also divide zero, but the word positive excludes zero from this minimum.

Suppose two events happen every twelve and eighteen days, and both happen at day $t=0$. Their next meeting is at day $36$, after three twelve-day intervals and two eighteen-day intervals. Through day $72$, their common event times are $0,36,72$.

The prime factorization explains why this coincidence occurs. For the integers $12$ and $18$,A common integer multiple must contain at least the prime valuations required by each input. Taking the larger prime valuation at each prime number meets both requirements with no extra factors. The greatest common divisor uses the smaller prime valuations instead. This is the same coordinate comparison developed under prime factorization, applied in opposite directions.

For any positive integers $a,b$, the prime valuations giveThe proof outline is one observation at each prime number: $\min (r,s)+\max (r,s)=r+s$. Hence the two sides have identical prime factorizations. Numerically, $6\cdot 36=12\cdot 18=216$. This also gives a calculation using the Euclidean algorithm for the greatest common divisor, without first finding the prime factors of both inputs.

Every positive common integer multiple is an integer multiple of the least common multiple, because its prime valuations must be at least the larger input prime valuations. Including negatives and zero, the common integer multiples of $12$ and $18$ are exactly $36k$ for integers $k$. For more than two positive integers, the same definition and rule of taking the largest prime valuations apply; one may also combine inputs successively. For example, $\mathrm{lcm}(12,18,20)=\mathrm{lcm}(36,20)=180$.

For nonzero signed integers, take absolute values: $\mathrm{lcm}(-12,18)=36$. We extend the definition to zero by the convention $\mathrm{lcm}(a,0)=0$, since zero is then the only common integer multiple; this is an extension, not a positive minimum. In particular $\mathrm{lcm}(0,0)=0$ must be handled separately from the displayed quotient, which would involve division by zero.

The least common multiple is useful for combining fractions with the smallest positive shared denominator. For example,Its scheduling interpretation requires care about starting times. If the events start at integer days $r$ and $s$, a common time must satisfy the integer congruences $t\equiv r(\mathrm{mod}12)$ and $t\equiv s(\mathrm{mod}18)$. The Chinese remainder theorem for noncoprime moduli says a solution exists exactly when $6\mid (r-s)$. If it exists, solutions form one congruence class modulo $36$, but the first meeting need not be day $36$. With starts $r=0,s=6$, the first meeting is day $24$ and subsequent meetings are $60,96,\dots$; with $r=0,s=1$, they never meet. The beauty of the least common multiple is the exact shared spacing, while the integer congruences determine whether and where the spacing begins.

Book VII, Proposition 34 of Euclid's Elements constructs the least common multiple of two numbers, and Proposition 36 extends the construction to three by first combining two of them. In modern notation this is $\mathrm{lcm}(a,b,c)=\mathrm{lcm}(\mathrm{lcm}(a,b),c)$. These are specific ancient treatments of the construction, rather than a claim about its first discovery. ^{[ref] [ref]}

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all of the given integers. In other words, the LCM is the smallest number that each of the integers can divide evenly into.