Greatest Common Divisor (GCD) of 140 and 81
The greatest common divisor (GCD) of 140 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 140 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 | 140 ÷ 81 = 1 remainder 59 |
| 2 | 81 ÷ 59 = 1 remainder 22 |
| 3 | 59 ÷ 22 = 2 remainder 15 |
| 4 | 22 ÷ 15 = 1 remainder 7 |
| 5 | 15 ÷ 7 = 2 remainder 1 |
| 6 | 7 ÷ 1 = 7 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 23 and 25 | 1 |
| 164 and 149 | 1 |
| 165 and 196 | 1 |
| 132 and 144 | 12 |
| 137 and 128 | 1 |