Greatest Common Divisor (GCD) of 78 and 123
The greatest common divisor (GCD) of 78 and 123 is 3.
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 123?
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 ÷ 123 = 0 remainder 78 |
| 2 | 123 ÷ 78 = 1 remainder 45 |
| 3 | 78 ÷ 45 = 1 remainder 33 |
| 4 | 45 ÷ 33 = 1 remainder 12 |
| 5 | 33 ÷ 12 = 2 remainder 9 |
| 6 | 12 ÷ 9 = 1 remainder 3 |
| 7 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 42 and 101 | 1 |
| 126 and 112 | 14 |
| 88 and 147 | 1 |
| 120 and 140 | 20 |
| 84 and 107 | 1 |