Greatest Common Divisor (GCD) of 188 and 105
The greatest common divisor (GCD) of 188 and 105 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 188 and 105?
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 | 188 ÷ 105 = 1 remainder 83 |
| 2 | 105 ÷ 83 = 1 remainder 22 |
| 3 | 83 ÷ 22 = 3 remainder 17 |
| 4 | 22 ÷ 17 = 1 remainder 5 |
| 5 | 17 ÷ 5 = 3 remainder 2 |
| 6 | 5 ÷ 2 = 2 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 63 and 96 | 3 |
| 52 and 112 | 4 |
| 63 and 53 | 1 |
| 143 and 52 | 13 |
| 98 and 189 | 7 |