Greatest Common Divisor (GCD) of 133 and 193
The greatest common divisor (GCD) of 133 and 193 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 133 and 193?
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 | 133 ÷ 193 = 0 remainder 133 |
| 2 | 193 ÷ 133 = 1 remainder 60 |
| 3 | 133 ÷ 60 = 2 remainder 13 |
| 4 | 60 ÷ 13 = 4 remainder 8 |
| 5 | 13 ÷ 8 = 1 remainder 5 |
| 6 | 8 ÷ 5 = 1 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 |
|---|---|
| 186 and 193 | 1 |
| 126 and 76 | 2 |
| 23 and 152 | 1 |
| 117 and 93 | 3 |
| 134 and 73 | 1 |