Greatest Common Divisor (GCD) of 109 and 81
The greatest common divisor (GCD) of 109 and 81 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 109 and 81?
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 | 109 ÷ 81 = 1 remainder 28 |
| 2 | 81 ÷ 28 = 2 remainder 25 |
| 3 | 28 ÷ 25 = 1 remainder 3 |
| 4 | 25 ÷ 3 = 8 remainder 1 |
| 5 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 105 and 21 | 21 |
| 142 and 11 | 1 |
| 140 and 153 | 1 |
| 91 and 100 | 1 |
| 49 and 151 | 1 |