Greatest Common Divisor (GCD) of 81 and 105
The greatest common divisor (GCD) of 81 and 105 is 3.
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 105?
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 ÷ 105 = 0 remainder 81 |
| 2 | 105 ÷ 81 = 1 remainder 24 |
| 3 | 81 ÷ 24 = 3 remainder 9 |
| 4 | 24 ÷ 9 = 2 remainder 6 |
| 5 | 9 ÷ 6 = 1 remainder 3 |
| 6 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 135 and 130 | 5 |
| 56 and 163 | 1 |
| 167 and 95 | 1 |
| 61 and 148 | 1 |
| 137 and 135 | 1 |