**Definition of Least Common Multiple (LCM)** **:**

The **least common multiple (LCM)** of two natural numbers *n* and *m* is the smallest valued number *h* so that *h* is a multiple of both *n* and *m*. We symbolize this as LCM(*n*, *m*) = *h*.

Example:

Relating this definition to the problem we solved on the previous page, we see that the two numbers *n* and *m* are the numbers 12 and 20. We may write the solution symbolically as LCM(12, 20) = 60. This says **"the least common multiple of 12 and 20 is 60."**

The number *h* in the definition corresponds to the 60 in the problem. It has the least value if the multiples shared by both *n* and *m.* The number *h* is the Least Common Multiple of 12 and 20.

One method for finding the least common multiple is to choose the value that is **the minimum value in the intersection of the sets of multiples.**

Example: Find the LCM(12, 20).

We find the set of their natural number multiples and then find the least value in the intersection of the two sets.

Set of multiples of 12 is *A* = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, …}

Set of multiples of 20 is *B* = {20, 40, 60, 80, 100, 120, 140, 160, 180, …}

*A* ∩ *B* = {60, 120, 180, …} which has the least value 60.

The LCM(12, 20) = 60.

Notice that the intermediate step in the problem gives all the common multiples.

Since we have found the least common multiple, 60, for the gear problem, we can now answer the question to the gear problem. The question asked how many revolutions are needed to realign the mark. Since we know the first gear must pass 60 teeth and after every 12 it has made one revolution, we know it must make 5 revolutions before the marks are aligned for the first time (60 ÷ 12 = 5). (Also, note that the second gear must make three revolutions for 60 teeth to pass the mark, 60 ÷ 20 = 3.)

As with the GCF, finding the LCM of larger numbers by listing multiples is not very practical. Also, just like the GCF, we can use **prime factorization** to help.

Example: Find LCM(308,1176).

Again we list the prime factors of the values.

308 = 2^{2} · 7 · 11

1176 = 2^{3} · 3 · 7^{2}

Since we are looking for multiples, every multiple of 308 must contain 2^{2} · 7 · 11 as factors and every multiple of 1176 must contain 2^{3} · 3 · 7^{2} as factors. This means that 2, 3, 7, and 11 must be prime factors in any common multiple. For 2 and 7, which appear in both lists, we need the highest power of that prime as a factor to create the LCM, since 2^{3} is automatically a multiple of 2^{2}. So using the highest power of each prime, LCM (308,1176) = 2^{3} · 3 · 7^{2} · 11 = 12,936.

**Entities should not be multiplied unnecessarily.**

—**William of Occam (1300-1439)**