Greatest Common Divisor (GCD) of 123 and 89
The greatest common divisor (GCD) of 123 and 89 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 123 and 89?
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 | 123 ÷ 89 = 1 remainder 34 |
| 2 | 89 ÷ 34 = 2 remainder 21 |
| 3 | 34 ÷ 21 = 1 remainder 13 |
| 4 | 21 ÷ 13 = 1 remainder 8 |
| 5 | 13 ÷ 8 = 1 remainder 5 |
| 6 | 8 ÷ 5 = 1 remainder 3 |
| 7 | 5 ÷ 3 = 1 remainder 2 |
| 8 | 3 ÷ 2 = 1 remainder 1 |
| 9 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 81 and 120 | 3 |
| 129 and 112 | 1 |
| 61 and 123 | 1 |
| 80 and 84 | 4 |
| 116 and 107 | 1 |