Greatest Common Divisor (GCD) of 34 and 128
The greatest common divisor (GCD) of 34 and 128 is 2.
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 34 and 128?
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 | 34 ÷ 128 = 0 remainder 34 |
| 2 | 128 ÷ 34 = 3 remainder 26 |
| 3 | 34 ÷ 26 = 1 remainder 8 |
| 4 | 26 ÷ 8 = 3 remainder 2 |
| 5 | 8 ÷ 2 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 195 and 52 | 13 |
| 169 and 63 | 1 |
| 136 and 147 | 1 |
| 49 and 196 | 49 |
| 129 and 88 | 1 |