Greatest Common Divisor (GCD) of 105 and 189
The greatest common divisor (GCD) of 105 and 189 is 21.
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 105 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 | 105 ÷ 189 = 0 remainder 105 |
| 2 | 189 ÷ 105 = 1 remainder 84 |
| 3 | 105 ÷ 84 = 1 remainder 21 |
| 4 | 84 ÷ 21 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 91 and 154 | 7 |
| 10 and 65 | 5 |
| 89 and 52 | 1 |
| 137 and 116 | 1 |
| 185 and 159 | 1 |