Greatest Common Divisor (GCD) of 120 and 189
The greatest common divisor (GCD) of 120 and 189 is 3.
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 120 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 | 120 ÷ 189 = 0 remainder 120 |
| 2 | 189 ÷ 120 = 1 remainder 69 |
| 3 | 120 ÷ 69 = 1 remainder 51 |
| 4 | 69 ÷ 51 = 1 remainder 18 |
| 5 | 51 ÷ 18 = 2 remainder 15 |
| 6 | 18 ÷ 15 = 1 remainder 3 |
| 7 | 15 ÷ 3 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 45 and 174 | 3 |
| 15 and 188 | 1 |
| 108 and 199 | 1 |
| 168 and 187 | 1 |
| 126 and 83 | 1 |