Greatest Common Divisor (GCD) of 51 and 180
The greatest common divisor (GCD) of 51 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 51 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 | 51 ÷ 180 = 0 remainder 51 |
| 2 | 180 ÷ 51 = 3 remainder 27 |
| 3 | 51 ÷ 27 = 1 remainder 24 |
| 4 | 27 ÷ 24 = 1 remainder 3 |
| 5 | 24 ÷ 3 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 102 and 66 | 6 |
| 55 and 137 | 1 |
| 72 and 71 | 1 |
| 200 and 165 | 5 |
| 59 and 108 | 1 |