The lowest common denominator is **the smallest number which all the numbers can be divided into**. The lowest common denominator is also called as least common multiple(lcm).

It is termed as **common factors x uncommon factors**

## How do you find the lowest common denominator?

The following are three methods to find the lowest denominator.

## 1. Lowest Common Denominator – Division Method

In this method divide, all the numbers till the remainder are 1. For example lcm of 6,8,9 and 12

Divisor | 6 | 8 | 9 | 12 |

2 | 3 | 4 | 9 | 6 |

3 | 1 | 4 | 3 | 2 |

2 | 1 | 2 | 3 | 1 |

2 | 1 | 1 | 3 | 1 |

3 | 1 | 1 | 1 | 1 |

- Find the factors of the first number, 6 = 2 x 3.
- Take the first factor and divide it by each. 6,8 and 12 when divided by 2 give 3,4 and 6 respectively. As 9 is not divisible by 2. Keeping 9 as it is
- Taking the other prime number 3 and dividing the numbers 3 and 6 which gives the division as 1 and 2. Repeat the process until every factor’s remainder is 1
- Multiply the divisors
**2 x 3 x 2 x 2 x 3 = 72**

So the lcm or least common denomination is **72**

## 2. Least Common Multiple – Prime Factorization Method

In the prime factorization method, we find the prime factors of the numbers. Considering the same example,

- factors are:

6 = 2 x 3

8 = 2 x 2 x 2

9 = 3 x 3

12 = 2 x 2 x3

2. Find common factor between all numbers, in this case it is **1**

3. Find common factors between maximum numbers.

The common factor between 6, 8, and 12 is 2

The common factor between 6, 9, and 12 is 3

4. Find the remaining uncommon factors from all the numbers:

uncommon factors in 6 are 1, in 8 is 2 x 2, 9 is 3, 12 is 1

5. Common factors x uncommon factors =** 2 x 3 x 2 x 2 x 3 = 72**

## 3. Listing the Multiples

Multiples of 6 | Multiples of 8 | Multiples of 9 | Multiples of 12 |

6 | 8 | 9 | 12 |

12 | 16 | 18 | 24 |

18 | 24 | 27 | 36 |

24 | 32 | 36 | 48 |

30 | 40 | 45 | 60 |

36 | 48 | 54 | 72 |

42 | 56 | 63 | 84 |

48 | 64 | 72 | 96 |

54 | 72 | 81 | 108 |

60 | 80 | 90 | 120 |

72 | 88 | 99 | 132 |

After listing the multiples the smallest common number between all the numbers is 72

therefore **lcm of 6, 8, 9, and 12 is 72**

## Questions below are taken from a book of an author R.S.Agarwal.

## FAQ: Detailed answer can be found in pdf below

## 1. Find H.C.F of numbers36 and 84

12

## 2.Find G.C.D of numbers204, 1190, 1445

17

## 3.Find GCD of 2923 and 3239

79

## 4.Find the LCM of numbers 24,36 and 40

360

## 5. Find LCM of numbers 22, 54, 108, 135 and 198

198

## 6. Find GCD of fractions 2/3, 8/9, 64/81, 10/27

2/3

## 7.Find HCF of fractions 9/10, 12/25, 18/35 and 21/40

3/5

## 8.Find LCM of fractions 1/3, 5/6, 2/9, 4/27

20/54 = 10/27

## 9. Find GCD of decimal numbers 1.08, 0.36 and 0.9

0.9

## 10. Find LCM of 3, 2.7 and 0.9

2.7

## 11. HCF of 3240, 3600 and a third number is 36 and their LCM is 2^4 x 3^5 x 5^2 x 7^2

2^2 x 3^5 x 7^2

## 12.Three numbers are in ratio 2:3 and their HCF is 12. What are the numbers?

12,24, 36

## 13. The sum of two numbers is 216 and their HCF is 27. Find the numbers.

27, 27×7

## 14. The sum of two numbers is 528 and their HCF is 33. The number of pairs of numbers satisfying above condition is ?

Total 4 pairs

## 15. The number of pairs lying between 40 and 100 with GCD 15 are …

Total 4 pairs

## 16.GCD of two numbers is 12 and their difference is 12. Find numbers

12×4, 12×3

## 17. The product of two numbers is 4107. The GCD of those numbers is 37. What is the greater number?

111

## 18.The product of two numbers is 2028 and their HCF is 13. The number of such pairs are how many

Total pairs are 3

## Download a Free pdf Sample of Examples(Questions from the book R.S. Agarwal)

You can find more examples here

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