Greatest Common Divisor (GCD) of 123 and 164
The greatest common divisor (GCD) of 123 and 164 is 41.
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 164?
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 ÷ 164 = 0 remainder 123 |
| 2 | 164 ÷ 123 = 1 remainder 41 |
| 3 | 123 ÷ 41 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 143 and 94 | 1 |
| 54 and 111 | 3 |
| 84 and 90 | 6 |
| 143 and 145 | 1 |
| 28 and 42 | 14 |