Greatest Common Divisor (GCD) of 41 and 68
The greatest common divisor (GCD) of 41 and 68 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 41 and 68?
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 | 41 ÷ 68 = 0 remainder 41 |
| 2 | 68 ÷ 41 = 1 remainder 27 |
| 3 | 41 ÷ 27 = 1 remainder 14 |
| 4 | 27 ÷ 14 = 1 remainder 13 |
| 5 | 14 ÷ 13 = 1 remainder 1 |
| 6 | 13 ÷ 1 = 13 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 111 and 105 | 3 |
| 185 and 80 | 5 |
| 78 and 45 | 3 |
| 133 and 199 | 1 |
| 59 and 122 | 1 |