Greatest Common Divisor (GCD) of 125 and 157
The greatest common divisor (GCD) of 125 and 157 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 125 and 157?
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 | 125 ÷ 157 = 0 remainder 125 |
| 2 | 157 ÷ 125 = 1 remainder 32 |
| 3 | 125 ÷ 32 = 3 remainder 29 |
| 4 | 32 ÷ 29 = 1 remainder 3 |
| 5 | 29 ÷ 3 = 9 remainder 2 |
| 6 | 3 ÷ 2 = 1 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 10 and 111 | 1 |
| 169 and 31 | 1 |
| 183 and 11 | 1 |
| 13 and 45 | 1 |
| 180 and 196 | 4 |