Greatest Common Divisor (GCD) of 125 and 199
The greatest common divisor (GCD) of 125 and 199 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 199?
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 ÷ 199 = 0 remainder 125 |
| 2 | 199 ÷ 125 = 1 remainder 74 |
| 3 | 125 ÷ 74 = 1 remainder 51 |
| 4 | 74 ÷ 51 = 1 remainder 23 |
| 5 | 51 ÷ 23 = 2 remainder 5 |
| 6 | 23 ÷ 5 = 4 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 185 | 1 |
| 88 and 152 | 8 |
| 25 and 58 | 1 |
| 154 and 45 | 1 |
| 180 and 81 | 9 |