Greatest Common Divisor (GCD) of 45 and 197
The greatest common divisor (GCD) of 45 and 197 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 45 and 197?
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 | 45 ÷ 197 = 0 remainder 45 |
| 2 | 197 ÷ 45 = 4 remainder 17 |
| 3 | 45 ÷ 17 = 2 remainder 11 |
| 4 | 17 ÷ 11 = 1 remainder 6 |
| 5 | 11 ÷ 6 = 1 remainder 5 |
| 6 | 6 ÷ 5 = 1 remainder 1 |
| 7 | 5 ÷ 1 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 198 and 51 | 3 |
| 13 and 87 | 1 |
| 15 and 69 | 3 |
| 136 and 110 | 2 |
| 161 and 198 | 1 |