Greatest Common Divisor (GCD) of 78 and 126
The greatest common divisor (GCD) of 78 and 126 is 6.
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 126?
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 ÷ 126 = 0 remainder 78 |
| 2 | 126 ÷ 78 = 1 remainder 48 |
| 3 | 78 ÷ 48 = 1 remainder 30 |
| 4 | 48 ÷ 30 = 1 remainder 18 |
| 5 | 30 ÷ 18 = 1 remainder 12 |
| 6 | 18 ÷ 12 = 1 remainder 6 |
| 7 | 12 ÷ 6 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 91 and 41 | 1 |
| 44 and 132 | 44 |
| 136 and 66 | 2 |
| 23 and 47 | 1 |
| 128 and 126 | 2 |