Greatest Common Divisor (GCD) of 103 and 188
The greatest common divisor (GCD) of 103 and 188 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 103 and 188?
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 | 103 ÷ 188 = 0 remainder 103 |
| 2 | 188 ÷ 103 = 1 remainder 85 |
| 3 | 103 ÷ 85 = 1 remainder 18 |
| 4 | 85 ÷ 18 = 4 remainder 13 |
| 5 | 18 ÷ 13 = 1 remainder 5 |
| 6 | 13 ÷ 5 = 2 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 |
|---|---|
| 134 and 54 | 2 |
| 78 and 184 | 2 |
| 186 and 105 | 3 |
| 36 and 125 | 1 |
| 97 and 63 | 1 |