Greatest Common Divisor (GCD) of 119 and 189
The greatest common divisor (GCD) of 119 and 189 is 7.
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 119 and 189?
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 | 119 ÷ 189 = 0 remainder 119 |
| 2 | 189 ÷ 119 = 1 remainder 70 |
| 3 | 119 ÷ 70 = 1 remainder 49 |
| 4 | 70 ÷ 49 = 1 remainder 21 |
| 5 | 49 ÷ 21 = 2 remainder 7 |
| 6 | 21 ÷ 7 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 161 and 125 | 1 |
| 173 and 126 | 1 |
| 140 and 149 | 1 |
| 162 and 174 | 6 |
| 25 and 113 | 1 |