Greatest Common Divisor (GCD) of 81 and 127
The greatest common divisor (GCD) of 81 and 127 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 127?
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 ÷ 127 = 0 remainder 81 |
| 2 | 127 ÷ 81 = 1 remainder 46 |
| 3 | 81 ÷ 46 = 1 remainder 35 |
| 4 | 46 ÷ 35 = 1 remainder 11 |
| 5 | 35 ÷ 11 = 3 remainder 2 |
| 6 | 11 ÷ 2 = 5 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 102 and 34 | 34 |
| 72 and 15 | 3 |
| 25 and 181 | 1 |
| 149 and 137 | 1 |
| 172 and 97 | 1 |