Greatest Common Divisor (GCD) of 65 and 90
The greatest common divisor (GCD) of 65 and 90 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 90?
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 ÷ 90 = 0 remainder 65 |
| 2 | 90 ÷ 65 = 1 remainder 25 |
| 3 | 65 ÷ 25 = 2 remainder 15 |
| 4 | 25 ÷ 15 = 1 remainder 10 |
| 5 | 15 ÷ 10 = 1 remainder 5 |
| 6 | 10 ÷ 5 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 51 and 86 | 1 |
| 20 and 197 | 1 |
| 185 and 32 | 1 |
| 195 and 62 | 1 |
| 68 and 181 | 1 |