Greatest Common Divisor (GCD) of 75 and 199
The greatest common divisor (GCD) of 75 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 75 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 | 75 ÷ 199 = 0 remainder 75 |
| 2 | 199 ÷ 75 = 2 remainder 49 |
| 3 | 75 ÷ 49 = 1 remainder 26 |
| 4 | 49 ÷ 26 = 1 remainder 23 |
| 5 | 26 ÷ 23 = 1 remainder 3 |
| 6 | 23 ÷ 3 = 7 remainder 2 |
| 7 | 3 ÷ 2 = 1 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 179 and 79 | 1 |
| 64 and 51 | 1 |
| 42 and 126 | 42 |
| 30 and 144 | 6 |
| 199 and 127 | 1 |