Greatest Common Divisor (GCD) of 71 and 194
The greatest common divisor (GCD) of 71 and 194 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 71 and 194?
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 | 71 ÷ 194 = 0 remainder 71 |
| 2 | 194 ÷ 71 = 2 remainder 52 |
| 3 | 71 ÷ 52 = 1 remainder 19 |
| 4 | 52 ÷ 19 = 2 remainder 14 |
| 5 | 19 ÷ 14 = 1 remainder 5 |
| 6 | 14 ÷ 5 = 2 remainder 4 |
| 7 | 5 ÷ 4 = 1 remainder 1 |
| 8 | 4 ÷ 1 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 42 and 105 | 21 |
| 137 and 162 | 1 |
| 126 and 22 | 2 |
| 48 and 72 | 24 |
| 193 and 141 | 1 |