Greatest Common Divisor (GCD) is the largest common factor of a number. It is also called as a greatest common factor (GCF).

## Finding the Greatest Common Divisor:

Say we have 3 numbers: 60, 132, and 96, and want to find the largest number that will divide evenly into each of these numbers. Two different methods can be used to find the solution to this problem.

#### Method I:

List the factors of each number and then identify the greatest number that is on every list.

**Factors of 60**: 1, 2, 3, 4, 5, 6, 10, **12**, 15, 20, 30, 60**Factors of 132**: 1, 2, 3, 4, 6, 11, **12**, 22, 33, 44, 66, 132**Factors of 96**: 1, 2, 3, 4, 6, 8, **12**, 16, 24, 32, 48. 96

So the greatest common factor would be **12**

#### Method II:

Write the prime factorization of each number. The GCF is the product of the common prime factors. Notice that the prime number 2 occurs twice in each of the three numbers. So we have determined that the greatest common factor (GCF) is 2 • 2 • 3 = 12.

## GCF of terms that include variable:

In algebra, sometimes you need to find the greatest common factor with numbers that include variables. The strategies for finding the answers are very straightforward.

For example, consider finding the greatest common factor of 16x^{2}y and 18xy^{2}. First, find the prime factorization for both numbers as well as the factors of the variables:

16x^{2}y = 2 • 2 • 2 • 2 • x • x • y

18xy^{2}= 2 • 3 • 3 • x • y •y

Both the terms have 2, x and y in common. Therefore the GCD or GCF is 2xy.

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

## 1. Find H.C.F of numbers 36 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 a ratio of 1:2:3 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 the above condition is?

Total 4 pairs

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

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 is how many

Total pairs are 3