## Least Common Multiple (LCM)

#### Result Summary:

 Type Result Given set of numbers 0 Least Common Multiple (LCM) 0

### What is LCM?

A multiple is a value which is obtained when a number is multiplied by another number. The multiples of any given number can be easily determined from its multiplication table. For example, the multiples of 3 are 3, 6, 9, 12, 15, … The multiples do not include 0.

In mathematics, for a given set of two or more numbers, the least common multiple (LCM), also known as the least common divisor (LCD) is the smallest value which is divisible by all the numbers. By common multiple, we mean a number which is a multiple of two or more numbers.

LCM is used to perform arithmetic operations on fractions when their denominators are different. If we need to add or subtract fractions, the LCM can be used to make the denominators common for all the fractions. This aids in the simplification process.

Consider the numbers 2, 3 and 4. We can find the LCM by listing out their multiples first:

2 = 2, 4, 6, 8, 10, 12, 14, …

3 = 3, 6, 9, 12, 15, 18, 21, …

4 = 4, 8, 12, 16, 20, 24, 28, …

Listing the multiples of the numbers clearly shows that 12 is the smallest number which is a multiple of these three numbers.

### Properties of LCM

Given below are some important properties of the LCM.

Multiple – The multiple of any number is a value obtained by multiplying that number with several other numbers. This can be easily observed in a multiplication table.

LCM – For a set of numbers, the LCM is the smallest number which is a multiple of all the numbers in the set.

Associative property – LCM (a, b) = LCM (b, a). For example, LCM (4, 5) = 20 and LCM (5, 4) = 20.

Commutative property – LCM (a, b, c) = LCM (LCM (a, b), c) = LCM (a, LCM (b, c)). For example, to find LCM (4, 5, 6), first compute LCM (LCM (4, 5), 6) = LCM (20, 6) = 60. Now, LCM (4, LCM (5, 6)) = LCM (4, 30) = 60.

Distributive property – LCM (ad, bd, cd) = d LCM (b, a). For example, LCM (8, 10, 12) = 120 and 2 × LCM (4, 5, 6) = 2 × 60 = 120.

### Finding the LCM

There are four ways to find the LCM of a given set of numbers.

1) Listing the multiples: The simplest way to find the LCM of some numbers, is to list out all the multiples and find the smallest multiple common to them.

For example, find the LCM of 20 and 90.

20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, ...

90 = 90, 180, 270, 360, …

Hence, the LCM is 180.

2) Prime factoring: To find the LCM, prime factoring is another method. In this case, list out only the prime factors of each number (even if they are repeated). Finally, multiply these prime factors as each occurs most often to obtain the LCM.

For example, find the LCM of 20, 60, 150.

20 = 2 × 2 × 5.

50 = 2 × 5 × 5.

150 = 2 × 3 × 5 × 5.

The highest occurrence of 2 is 2, of 3 is 1, and of 5 is 2.

Hence, the LCM = 2 × 2 × 3 × 5 × 5 = 300.

3) LCM by division:

i) In this method, we start by writing the numbers on top in a row.

ii) Now we divide each number in the row by the smallest prime number. If any number is not divisible by the prime number, we simply put down that number and proceed.

iii) Keep on dividing the numbers in each row by more prime numbers until all the numbers in a row are 1.

iv) The LCM is the product of the prime numbers written in the first column.

Find the LCM of 10, 15, 20.

Division table

 10 15 20 2 5 15 10 3 5 5 10 5 1 1 2 2 1 1 1

LCM = product of all the numbers in the first column = 2 × 3 × 5 × 2 = 60

4) LCM from the GCF: To obtain the LCM from the GCF, the formula is

LCM(a,b) = (a×b) / (GCF(a,b))

Find LCM (15, 20)

Now, the GCF (15, 20) = 5.

Hence, LCM(15,20)=(15×20)/5=60

### Characteristics of LCM

• The LCM of a set of numbers cannot be less than any of them. The LCM is always greater than the numbers involved.
• If one of the numbers in the set is a factor of another factor, the LCM is the bigger number. For example, the LCM (2, 4) = 4.
• For any two numbers, the product of the GCF and LCM is equal to the product of the numbers itself.
• The LCM can be defined only for positive integers.
• If the numbers are prime to each other, then the LCM is simply the product of the numbers. For example, LCM (3, 5, 7) = 3 × 5 × 7 = 105.

• ### Areas of application

The LCM is frequently used in the following situations:

• To analyse an event which is repeating.
• To obtain multiple items so that we have enough.
• To figure out how and when an event will happen again.
• Given below is a real-life use of LCM.

Question: Bell A rings every 18 seconds, and bell B rings every 45 seconds. At 5:00 pm, the two bells ring simultaneously. At what time will the bells ring again at the same time?

Answer: To do this problem, we just need to find the LCM (18, 45). The prime factoring method is used here.

18 = 2 × 3 × 3, 45 = 5 × 3 × 3.

Here, 3 occurs a maximum of 2 times, 2 appears once and 5 appears once.

Hence, the LCM (18, 45) = 2 × 3 × 3 × 5 = 90 seconds = 1.5 minutes.

So, the bells will ring together again at 5:01:30 pm.