Greatest Common Divisor (GCD) of 21 and 23
The greatest common divisor (GCD) of 21 and 23 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 23?
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 ÷ 23 = 0 remainder 21 |
| 2 | 23 ÷ 21 = 1 remainder 2 |
| 3 | 21 ÷ 2 = 10 remainder 1 |
| 4 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 193 and 84 | 1 |
| 145 and 141 | 1 |
| 193 and 38 | 1 |
| 79 and 194 | 1 |
| 180 and 155 | 5 |