Greatest Common Divisor (GCD) of 89 and 140
The greatest common divisor (GCD) of 89 and 140 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 89 and 140?
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 | 89 ÷ 140 = 0 remainder 89 |
| 2 | 140 ÷ 89 = 1 remainder 51 |
| 3 | 89 ÷ 51 = 1 remainder 38 |
| 4 | 51 ÷ 38 = 1 remainder 13 |
| 5 | 38 ÷ 13 = 2 remainder 12 |
| 6 | 13 ÷ 12 = 1 remainder 1 |
| 7 | 12 ÷ 1 = 12 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 47 and 94 | 47 |
| 103 and 69 | 1 |
| 145 and 21 | 1 |
| 197 and 98 | 1 |
| 181 and 165 | 1 |