Greatest Common Divisor (GCD) of 78 and 147
The greatest common divisor (GCD) of 78 and 147 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 147?
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 ÷ 147 = 0 remainder 78 |
| 2 | 147 ÷ 78 = 1 remainder 69 |
| 3 | 78 ÷ 69 = 1 remainder 9 |
| 4 | 69 ÷ 9 = 7 remainder 6 |
| 5 | 9 ÷ 6 = 1 remainder 3 |
| 6 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 85 and 140 | 5 |
| 121 and 137 | 1 |
| 107 and 27 | 1 |
| 81 and 171 | 9 |
| 160 and 154 | 2 |