Greatest Common Divisor (GCD) of 119 and 68
The greatest common divisor (GCD) of 119 and 68 is 17.
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 119 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 | 119 ÷ 68 = 1 remainder 51 |
| 2 | 68 ÷ 51 = 1 remainder 17 |
| 3 | 51 ÷ 17 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 116 and 86 | 2 |
| 189 and 19 | 1 |
| 128 and 46 | 2 |
| 176 and 132 | 44 |
| 70 and 191 | 1 |