Greatest Common Divisor (GCD) of 81 and 109
The greatest common divisor (GCD) of 81 and 109 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 81 and 109?
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 | 81 ÷ 109 = 0 remainder 81 |
| 2 | 109 ÷ 81 = 1 remainder 28 |
| 3 | 81 ÷ 28 = 2 remainder 25 |
| 4 | 28 ÷ 25 = 1 remainder 3 |
| 5 | 25 ÷ 3 = 8 remainder 1 |
| 6 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 138 and 45 | 3 |
| 148 and 200 | 4 |
| 72 and 98 | 2 |
| 176 and 77 | 11 |
| 128 and 192 | 64 |