Greatest Common Divisor (GCD) of 67 and 139
The greatest common divisor (GCD) of 67 and 139 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 139?
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 ÷ 139 = 0 remainder 67 |
| 2 | 139 ÷ 67 = 2 remainder 5 |
| 3 | 67 ÷ 5 = 13 remainder 2 |
| 4 | 5 ÷ 2 = 2 remainder 1 |
| 5 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 105 and 123 | 3 |
| 100 and 10 | 10 |
| 124 and 193 | 1 |
| 145 and 87 | 29 |
| 78 and 45 | 3 |