Greatest Common Divisor (GCD) of 65 and 106
The greatest common divisor (GCD) of 65 and 106 is 1.
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 106?
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 ÷ 106 = 0 remainder 65 |
| 2 | 106 ÷ 65 = 1 remainder 41 |
| 3 | 65 ÷ 41 = 1 remainder 24 |
| 4 | 41 ÷ 24 = 1 remainder 17 |
| 5 | 24 ÷ 17 = 1 remainder 7 |
| 6 | 17 ÷ 7 = 2 remainder 3 |
| 7 | 7 ÷ 3 = 2 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 143 and 139 | 1 |
| 52 and 67 | 1 |
| 104 and 35 | 1 |
| 86 and 171 | 1 |
| 102 and 25 | 1 |