Greatest Common Divisor (GCD) of 168 and 199
The greatest common divisor (GCD) of 168 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 168 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 | 168 ÷ 199 = 0 remainder 168 |
| 2 | 199 ÷ 168 = 1 remainder 31 |
| 3 | 168 ÷ 31 = 5 remainder 13 |
| 4 | 31 ÷ 13 = 2 remainder 5 |
| 5 | 13 ÷ 5 = 2 remainder 3 |
| 6 | 5 ÷ 3 = 1 remainder 2 |
| 7 | 3 ÷ 2 = 1 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 194 and 44 | 2 |
| 58 and 115 | 1 |
| 164 and 82 | 82 |
| 198 and 45 | 9 |
| 14 and 103 | 1 |