Greatest Common Divisor (GCD) of 67 and 108
The greatest common divisor (GCD) of 67 and 108 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 67 and 108?
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 | 67 ÷ 108 = 0 remainder 67 |
| 2 | 108 ÷ 67 = 1 remainder 41 |
| 3 | 67 ÷ 41 = 1 remainder 26 |
| 4 | 41 ÷ 26 = 1 remainder 15 |
| 5 | 26 ÷ 15 = 1 remainder 11 |
| 6 | 15 ÷ 11 = 1 remainder 4 |
| 7 | 11 ÷ 4 = 2 remainder 3 |
| 8 | 4 ÷ 3 = 1 remainder 1 |
| 9 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 160 and 70 | 10 |
| 199 and 71 | 1 |
| 54 and 69 | 3 |
| 146 and 111 | 1 |
| 146 and 166 | 2 |