Greatest Common Divisor (GCD) of 65 and 180
The greatest common divisor (GCD) of 65 and 180 is 5.
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 65 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 | 65 ÷ 180 = 0 remainder 65 |
| 2 | 180 ÷ 65 = 2 remainder 50 |
| 3 | 65 ÷ 50 = 1 remainder 15 |
| 4 | 50 ÷ 15 = 3 remainder 5 |
| 5 | 15 ÷ 5 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 157 and 65 | 1 |
| 79 and 144 | 1 |
| 124 and 39 | 1 |
| 168 and 103 | 1 |
| 43 and 164 | 1 |