Greatest Common Divisor (GCD) of 78 and 140
The greatest common divisor (GCD) of 78 and 140 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 140?
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 ÷ 140 = 0 remainder 78 |
| 2 | 140 ÷ 78 = 1 remainder 62 |
| 3 | 78 ÷ 62 = 1 remainder 16 |
| 4 | 62 ÷ 16 = 3 remainder 14 |
| 5 | 16 ÷ 14 = 1 remainder 2 |
| 6 | 14 ÷ 2 = 7 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 172 and 35 | 1 |
| 138 and 96 | 6 |
| 147 and 40 | 1 |
| 109 and 173 | 1 |
| 17 and 155 | 1 |