## 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 the 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.

**Application**

The dimension of using LCM of two numbers starts with basic math operations such as addition and subtraction on fractional numbers. In math problems where we pair two objects against each other, the LCM value is useful in optimizing the quantities of the given objects. Also, in computer science, the LCM of numbers helps design encoded messages using cryptography.

Fun Facts1. Greeks used the wax tablets to record multiplication tables in 1st century AD2. William Oughtred used the symbol “?” for multiplication in the 15th century to teach math |

## Related Math Vocabulary

- Multiples and multiplication
- Highest Common Factor (HCF)
- Fractions

## Practice Problems

## Least Common Multiple## 1LCM of two numbers is 8. If one of the numbers is 2, what could the other number be?8 16 10 5 CorrectIncorrect Correct answer is: 8 The LCM of a prime number and any of its multiples is the multiple itself. ## 2Identify the pair of numbers that have their LCM = 11.2 and 22 1 and 11 0 and 11 11 and 22 CorrectIncorrect Correct answer is: 1 and 11 The LCM of two numbers can only be a prime number if one of the numbers is the LCM itself and the other number is 1. ## 3Find the LCM of the numbers 16 and 20.320 20 16 80 CorrectIncorrect Correct answer is: 80 Prime factorization of 16 = 2 × 2 × 2 × 2 Prime factorization of 20 = 2 × 2 × 5 Pairing common factors and multiplying with independent factors, LCM= 2 × 2 × 2 × 2 × 5 = 80 ## 4Find the LCM of the numbers 2 and 5.2 5 10 7 CorrectIncorrect Correct answer is: 10 The lowest common multiple among the multiples of 2 and 5 is 10. Also, since 2 and 5 are prime numbers, their LCM is 2 × 5 = 10. |

## Frequently Asked Questions

**What is LCM?**

For a given set of numbers, the LCM is the smallest number that is a multiple of every one of the numbers.

**How do you find the LCM of two or more numbers?**

Start making multiples of each of them… one times the numbers, two times the numbers, three times the numbers, and so on. As soon as you find a multiple that is the same for all the numbers; that’s the least common multiple.

**What are LCM and GCF?**

The LCM of two or more numbers is the smallest number that is a multiple of every one of the numbers. The GCF of two or more numbers is the greatest factor that these numbers share.

**What is an application of LCM in real life?**

In music studios where different instruments are playing a different number of beats per minute (e.g., 10 bpm, 8 bpm, 2 bpm), music composers need to calculate the LCM to find out how many beats per minute they need.