How to Find The Least Common Multiple ( L.C.M. ) the Easy Way, and see some of its Applications

Step 1

Given the following three numbers, 3,4,and 6, we are interested in the smallest number that is divisible by 3,4 and 6. If we should find the product of all three numbers that is, we multiply (3)(4)(6) = 72, it seems logical that 72 should be easily divisible by 3,4, and 6 since 72 is the Product of these numbers. The BIG question is, "is 72 the smallest number that is divisible by each of these three numbers?". The answer is NO!, since 36, which is one-half of 72, is also divisible by each of the three numbers 3,4,and 6.

Step 2

The product 72 of 3,4, and 6, and the multiple 36 are called Common Multiples for the numbers 3,4, and 6. When any Number is used to multiply the set of Natural Numbers 1,2,3,4,5,... the products of that number with each of the natural numbers are called the multiples of that number. That is the Multiples of 3 are: 3(1),3(2),3(3),3(4),3(5),
3(6),... which are respectively equal to 3,6,9,12,15,18,... .

Step 3

The Multiples of 3 are: 3,6,9,12,15,18,... .
The Multiples of 4 are: 4,8,12,16,20,24,... .
The multiples of 6 are: 6,12,18,24,30,36,... .
Looking carefully at the three lists of Multiples for 3,4, and 6, we see that the Smallest Multiple that is common in each of the list is 12. We call 12 the Least Common Multiple or the L.C.M.

Step 4

There are several methods for finding the L.C.M. that may be easier than the method given in Step #3. An easy Method is; to take the three numbers 3,4, and 6, and begin by dividing each number by the smallest prime number that any of the numbers are divisible by, writing down the answer (quotient) of that division. If any of the numbers are not divisible by that prime number, simply write the number down. We continue this process until we have all 1's as our quotients, we then multiply all the prime divisors and their product will be our L.C.M. Please see image below.

Step 5

Let us look at a problem that involves finding the Least Common Denominator (L.C.M.) For example, in the supermarket, hotdog meat comes 8 in a package, hotdog buns come 12 in a package, hamburger meat comes 9 in a package and hamburger buns come 10 in a package. How many packages must you buy so that each hotdog meat and hamburger meat can be
put with its respective hotdog bun and hamburger bun? (there must not be any left over). To solve this problem, we find the L.C.M. of 8,9,10 and 12. Please see image below.

TIPS: