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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