Greatest Common Divisor (GCD) of 21 and 180
The greatest common divisor (GCD) of 21 and 180 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 21 and 180?
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 ÷ 180 = 0 remainder 21 |
| 2 | 180 ÷ 21 = 8 remainder 12 |
| 3 | 21 ÷ 12 = 1 remainder 9 |
| 4 | 12 ÷ 9 = 1 remainder 3 |
| 5 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 147 and 136 | 1 |
| 131 and 44 | 1 |
| 94 and 162 | 2 |
| 79 and 43 | 1 |
| 116 and 82 | 2 |