Greatest Common Divisor (GCD) of 109 and 68
The greatest common divisor (GCD) of 109 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 109 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 | 109 ÷ 68 = 1 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 |
|---|---|
| 128 and 33 | 1 |
| 144 and 46 | 2 |
| 108 and 108 | 108 |
| 135 and 88 | 1 |
| 122 and 45 | 1 |