Greatest Common Divisor (GCD) of 189 and 110
The greatest common divisor (GCD) of 189 and 110 is 1.
What is the Greatest Common Divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is useful in simplifying fractions, finding common factors, and in number theory.
How to Calculate the GCD of 189 and 110?
We use the Euclidean algorithm, which involves the following steps:
- Divide the larger number by the smaller number.
- Replace the larger number with the smaller number and the smaller number with the remainder from the division.
- Repeat this process until the remainder is zero.
- The non-zero remainder just before zero is the GCD.
Step-by-Step Euclidean Algorithm
| Step | Calculation |
|---|---|
| 1 | 189 ÷ 110 = 1 remainder 79 |
| 2 | 110 ÷ 79 = 1 remainder 31 |
| 3 | 79 ÷ 31 = 2 remainder 17 |
| 4 | 31 ÷ 17 = 1 remainder 14 |
| 5 | 17 ÷ 14 = 1 remainder 3 |
| 6 | 14 ÷ 3 = 4 remainder 2 |
| 7 | 3 ÷ 2 = 1 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 194 and 84 | 2 |
| 178 and 36 | 2 |
| 157 and 82 | 1 |
| 103 and 82 | 1 |
| 21 and 147 | 21 |