Algebra

# The Least Common Multiple Function

Last time, we discussed the Greatest Common Denominator function. Now, I turn to its counterpart, the LCM. What does the LCM function mean, and how can we calculate it?

LCM: A simple explanation

The LCM function, written as lcm(a, b), finds the lowest possible number into which both a and b will divide evenly. For example, the LCM of 6 and 8 is 24 because 24 is the smallest number that is divisible by 6 and 8.

Find the least common multiple of the following pairs of numbers: 3 and 6, 9 and 12, 4 and 5.

Notice that in the last example above, we arrive at a special case when a and b are relatively prime; that is, there is no number that divides evenly into both a and b except 1. In this case, the LCM turns out to be the product of the two numbers.

LCM: A concrete example

In the last post, I gave the example of tiling a rectangular floor with square tiles to illustrate the Greatest Common Denominator. In accordance with the reciprocity of these two functions, I will now present a similar yet opposite scenario.

Consider a large, elaborately patterned rectangular tile—say its dimensions are 4 feet by 6 feet—squatting unceremoniously on an unfinished concrete floor. Mr. Baxter the contractor examines this tile along with his new clients, Mr. and Mrs. Calliope. “So you see, Mr. Baxter,” young Mr. Calliope is saying in a slightly stilted dialect of English, “we must use this exact type of tile. No other tile will do, you see. But also, you see, this room must also be square, yes? Now you see the challenge.”

Mr. Baxter remains silent. Then, like a long-forgotten arcade machine, he forlornly spits out a terse reply. “Twelve feet.” Mr. and Mrs. Calliope, who have evidently forgotten their elementary arithmetic, cast him a brief look of distaste and turn back to the masterpiece of a floor tile. “Twelve feet, I reckon,” continues Mr. Baxter, straightening his hat as though it gave him courage to keep talking. “The room will have to be twelve feet square.”

At this, Mrs. Calliope finally registers what Mr. Baxter is saying. “Ah, you see!” she exclaims. “But how, yes, how can you be so sure, Signor?”

As you may have guessed, our contrary proposition for the lowest common multiple function, in contrast to tiling a rectangular floor with square tiles, is tiling a square floor with rectangular tiles. The task is to find the smallest possible square room into which our ornate rectangular tile will fit EXACTLY.

Find the smallest room that can be tiled evenly with the following sizes of tiles: 6 by 9, 2 by 4, 8 by 10.

LCM: Two methods for finding it

In the Greatest Common Denominator post, we discussed using prime factorization to find the GCF of two numbers. Continuing our contrast, we can use a similar yet opposite method to find the LCM of two numbers.

Consider the example lcm(20, 22). The prime factorization of 20 (which you can find using a factor tree) is $5\times2^2$. The prime factorization of 22 is $11\times2$.

In the Greatest Common Denominator post, we took each factor with the lowest power, and omitted it if it wasn’t in both a and b. For the LCM, we’ll take each factor with the highest power, and we’ll keep it even if it isn’t in both numbers. For instance, we’ll take 5, 2 squared because its power is higher than just 2, and 11. Multiplying our new list of factors, we get an LCM of 220.

For the second method, let’s look at a close opposite to the floor tiling method we examined in the GCF post. Then, we subtracted square regions until all that was left was a square region. Now, we add rectangular regions until we have a square region. For instance, given lcm(16, 20), 16 is less so we add 16 to itself. Then between 32 and 20, we add 20 to itself because 20 is less than 32. We then have 32 and 40, 48 and 40, 48 and 60, and 60 and 60. The numbers are now equal, so 60 is the LCM.

LCM: Applications

The LCM is used in situations where, like Mr. Baxter’s predicament, we need to find where two dimensions become equal. For instance, one would use lcm(15, 20) to find out the time between two trains meeting in a station. In mathematics, the LCM is crucial to adding and subtracting fractions. For instance, the denominator of 1/6 + 1/8 is lcm(6, 8).

Try these problems using LCM. Remember that the LCM of three numbers a, b, and c is simply lcm(lcm(a, b), c).

1. Two trains leave a station at 12:00 noon. The A train visits the station every 25 minutes, and the B train visits the station every 15 minutes. At what time will the two trains meet in the station again for the first time?

2. What is the smallest integer that is a multiple of 14, 9, and 21?