Greatest Common Divisor (GCD) of 90 and 189
The greatest common divisor (GCD) of 90 and 189 is 9.
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 90 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 | 90 ÷ 189 = 0 remainder 90 |
| 2 | 189 ÷ 90 = 2 remainder 9 |
| 3 | 90 ÷ 9 = 10 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 194 and 109 | 1 |
| 12 and 156 | 12 |
| 102 and 62 | 2 |
| 200 and 157 | 1 |
| 65 and 192 | 1 |