 # Least Common Multiple (LCM) Study Notes, Questions and Answers

Multiples are products of natural numbers.
Look at the multiplications of 3 below:
1 x 3 = 3
2 x 3 = 6
3 x 3 = 9
4 x 3 = 12
5 x 3 = 15
6 x 3 = 18
7 x 3 = 21
8 x 3 = 24
Hence, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24
Look at the multiplications of 4 below:
1 x 4 = 4
2 x 4 = 8
3 x 4 = 12
4 x 4 = 16
5 x 4 = 20
6 x 4 = 24
Hence, the multiples of 4 are 4, 8, 12, 16, 20, 24
The common multiples of 3 and 4 shown above are 12 and 24. Therefore the least common multiple of 3 and 4 is 12.

### Finding the LCM using Tables

For the GCD, the divisor (factor) has to be common to use it. For the LCM, if the smallest factor does not divide a number in the table, the number is retained. For example, we can find the LCM of 8, 12, 18, and 20 using tables as shown in the table below: To get the LCM, we multiply the prime factors; 2 x 2 x 2 x 3 x 3 x 5 = 23 x 32 x 5 = 360

In the first row, 8 ÷ 2 = 4, 12 ÷ 2 = 6, 18 ÷ 2 = 9, and 20 ÷ 2 = 10,
In the second row, 4 ÷ 2 = 2, 6 ÷ 2 = 3, and 10 ÷ 2 = 5. However, 9 is not divisible by 2, therefore we retain it as it is; 9.
When using tables, always use the least prime factor, in the example below, its 2. As demonstrated above, the factor 2 is followed by 3 and then 5.

### Finding the LCM: Practice Example 1

1. What is the LCM of 48, 100, and 72.

Method 1: Using Prime Factorization:
48: 2 x 2 x 2 x 2 x 3 = 24 x 31
100: 2 x 2 x 5 x 5 = 22 x 52
72: 2 x 2 x 2 x 3 x 3 = 23 x 32

Now, to find the LCM, you take the highest power of each prime factor: The highest power of 2 that appears is 24.
The highest power of 3 that appears is 32.
The highest power of 5 that appears is 52.
Multiply these highest powers together to find the LCM:
LCM = 4 x 32 x 52
LCM = 16 x 9 x 25
LCM = 3600

So, the LCM of 48, 100, and 72 is 3600.

Method 2: Using Tables: To get the LCM, we multiply the prime factors; 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5= 24 x 32 x 52 = 3600

## Notes and Questions Related to Least Common Multiple (LCM)

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