Greatest Common Divisor (GCD) of 77 and 140
The greatest common divisor (GCD) of 77 and 140 is 7.
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 77 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 | 77 ÷ 140 = 0 remainder 77 |
| 2 | 140 ÷ 77 = 1 remainder 63 |
| 3 | 77 ÷ 63 = 1 remainder 14 |
| 4 | 63 ÷ 14 = 4 remainder 7 |
| 5 | 14 ÷ 7 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 100 and 200 | 100 |
| 21 and 200 | 1 |
| 109 and 24 | 1 |
| 160 and 101 | 1 |
| 132 and 198 | 66 |