Greatest Common Divisor (GCD) of 113 and 194
The greatest common divisor (GCD) of 113 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 113 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 | 113 ÷ 194 = 0 remainder 113 |
| 2 | 194 ÷ 113 = 1 remainder 81 |
| 3 | 113 ÷ 81 = 1 remainder 32 |
| 4 | 81 ÷ 32 = 2 remainder 17 |
| 5 | 32 ÷ 17 = 1 remainder 15 |
| 6 | 17 ÷ 15 = 1 remainder 2 |
| 7 | 15 ÷ 2 = 7 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 35 and 38 | 1 |
| 165 and 83 | 1 |
| 138 and 135 | 3 |
| 144 and 64 | 16 |
| 69 and 100 | 1 |