Greatest Common Divisor (GCD) of 65 and 156
The greatest common divisor (GCD) of 65 and 156 is 13.
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 156?
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 ÷ 156 = 0 remainder 65 |
| 2 | 156 ÷ 65 = 2 remainder 26 |
| 3 | 65 ÷ 26 = 2 remainder 13 |
| 4 | 26 ÷ 13 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 130 and 135 | 5 |
| 182 and 105 | 7 |
| 150 and 147 | 3 |
| 170 and 126 | 2 |
| 110 and 51 | 1 |