Greatest Common Divisor (GCD) of 87 and 192
The greatest common divisor (GCD) of 87 and 192 is 3.
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 87 and 192?
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 | 87 ÷ 192 = 0 remainder 87 |
| 2 | 192 ÷ 87 = 2 remainder 18 |
| 3 | 87 ÷ 18 = 4 remainder 15 |
| 4 | 18 ÷ 15 = 1 remainder 3 |
| 5 | 15 ÷ 3 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 186 and 141 | 3 |
| 69 and 68 | 1 |
| 145 and 188 | 1 |
| 105 and 86 | 1 |
| 198 and 23 | 1 |