Least Common Multiple
Multiple is a number that can be divided by the given number without leaving a reminder. For example:
20 is a multiple of 5
Or, 5 × 4 = 20
And, 20 ÷ 5 = 4
Least common multiple
The least common multiple of two numbers is the “smallest non-zero common number” which is a multiple of both the numbers.
The different methods to find least common multiple of two or more numbers are:
- Using prime factorization
- Using repeated division
- Using multiples
1. LCM using prime factorization
In this method, a factorization tree for each given number is generated by listing the multiples of that number. The last branch of the tree has the least prime factors for that number.
For example, the factorization trees for 36 and 48 are generated as follows:
Figure: Prime factorization trees for the number 36 and 48
To find the LCM, pair the common multiples as shown. List them along with the remaining multiples.
LCM = 2 × 2 × 3 × 3 × 2 × 2
LCM = 144
2. LCM using repeated division
In this method, the given numbers are divided by the common divisors until there is no possible further division by the common number. The divisors and the remainders are multiplied together to obtain the LCM.
LCM = 2 × 2 × 3 × 4 × 3
LCM = 144
3. LCM using multiples
To find the LCM using multiples, list the multiples of the numbers in the table as shown. The least common multiple is the first common multiple for the given numbers.
1
2
3
4
5
6
7
8
9
10
11
12
36
36
72
108
144
180
216
252
288
324
360
396
432
48
48
96
144
192
240
288
336
384
432
480
528
576
For 36 and 48, the number 144 is the LCM.
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
| |
36
|
36
|
72
|
108
|
144
|
180
|
216
|
252
|
288
|
324
|
360
|
396
|
432
|
48
|
48
|
96
|
144
|
192
|
240
|
288
|
336
|
384
|
432
|
480
|
528
|
576
|
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