Least common multiple (LCM)

If one number is divisible by another without a remainder, then the first number is said to be a multiple of the second. The least common multiple (LCM) of several numbers is the smallest natural number that is divisible by these numbers without a remainder. In other words, the LCM of several numbers is the smallest natural number that is a multiple of these numbers.

How to find the LCM of two numbers

To find the LCM of two numbers, you need to
1) factorize these numbers into prime factors;
2) choose one of these numbers and write out all the factors included in the factorization of this number;
3) we add up those factors from the factorization of the second number that are not in the factorization of the selected number;
4) find the product of the resulting multipliers.

How to find the LCM of several numbers

To find the LCM of several numbers, you need to
1) factorize these numbers into prime factors;
2) choose one of these numbers and write out all the factors included in the factorization of this number;
3) sort through the remaining numbers in order, adding the missing factors from the factorization of other numbers;
4) find the product of the resulting multipliers.

The LCM of several numbers, just like the GCD, can be found gradually. First, the LCM of the first two numbers, then the LCM of the result and the third number, then the LCM of the result and the fourth number, etc.

Examples.

Example 1.
Find the least common multiple (LCM) of numbers 12 and 16.
Solution.
12 = 2*2*3;

16 = 2*2*2*2.

We choose the number 16. From the factorization of the number 12 we add to its factors the factor 3, which the number 16 does not have: 2*2*2*2*3=48.
Therefore, LCM(12, 16) = 48.
If you choose the number 12, then from the factorization of the number 16 you need to add the factor 2 twice, since in the number 12 there are only two factors of 2, and in the number 16 there are four: 2*2*3*2*2=48.
Therefore, LCM(12, 16) = 48.
Answer: 48.

Example 2.
Find the least common multiple (LCM) of numbers 396 and 180.
Solution.
396 = 2*2*3*3*11;

180 = 2*2*3*3*5.

We choose the number 396. From the factorization of the number 180 we add to its factors the factor 5, which the number 396 does not have. That is, 2*2*3*3*11*5=1980.
Therefore, LCM(396, 180) = 1980.
Answer: 1980.

Example 3.
Find the least common multiple (LCM) of numbers 34, 51 and 68.
Solution.
34 = 2*17;

51 = 3*17;

68 = 2*2*17.

We choose the number 34. From the factorization of the number 51 we add to its factors the factor 3, which the number 34 does not have. We obtain the factorization 2*17*3. From the factorization of the number 68 we add to its factors the factor 2, since in the number 68 there are two factors of 2, and in our expansion there is one factor of 2. That is, 2*17*3*2=204.
Therefore, LCM(34, 51, 68) = 204.
Answer: 204.

Example 4.
Find the least common multiple (LCM) of numbers 10, 21 and 36.
Solution.
10 = 2*5;

21 = 3*7;

36 = 2*2*3*3.

We choose the number 10. From the factorization of the number 21 we add to its factors the factors 3 and 7, which the number 10 does not have. We obtain the factorization 2*5*3*7. From the factorization of the number 36 we add to its factors the factors 2 and 3, since in the number 36 there are two factors of 2 and two factors of 3, and in our factorization there is one factor of 2 and one factor of 3. That is, 2*5*3*7*2*3=1260.
Therefore, LCM(10, 21, 36) = 1260.
Answer: 1260.

Example 5.
Find the least common multiple (LCM) of numbers 12, 30, 60 and 75.
Solution.
12 = 2*2*3;

30 = 2*3*5;

60 = 2*2*3*5;

75 = 3*5*5.

We choose the number 12. From the factorization of the number 30 we add to its factors the factor 5, which the number 12 does not have. We obtain the factorization 2*2*3*5. From the factorization of the number 60 we don't add anything, since all its factors are present in our factorization. From the factorization of the number 75 we add to its factors the factor 5, since in the number 75 there are two factors of 5, and in our factorization there is one factor of 5. That is, 2*2*3*5*5=300.
Therefore, LCM(12, 30, 60, 75) = 300.
Answer: 300.