Greatest Common Divisor (GCD) of 21 and 185
The greatest common divisor (GCD) of 21 and 185 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 21 and 185?
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 | 21 ÷ 185 = 0 remainder 21 |
| 2 | 185 ÷ 21 = 8 remainder 17 |
| 3 | 21 ÷ 17 = 1 remainder 4 |
| 4 | 17 ÷ 4 = 4 remainder 1 |
| 5 | 4 ÷ 1 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 170 and 118 | 2 |
| 28 and 74 | 2 |
| 117 and 117 | 117 |
| 175 and 194 | 1 |
| 45 and 46 | 1 |