Greatest Common Divisor (GCD) of 78 and 128
The greatest common divisor (GCD) of 78 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 78 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 | 78 ÷ 128 = 0 remainder 78 |
| 2 | 128 ÷ 78 = 1 remainder 50 |
| 3 | 78 ÷ 50 = 1 remainder 28 |
| 4 | 50 ÷ 28 = 1 remainder 22 |
| 5 | 28 ÷ 22 = 1 remainder 6 |
| 6 | 22 ÷ 6 = 3 remainder 4 |
| 7 | 6 ÷ 4 = 1 remainder 2 |
| 8 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 185 and 147 | 1 |
| 153 and 79 | 1 |
| 57 and 96 | 3 |
| 173 and 136 | 1 |
| 178 and 188 | 2 |