Greatest Common Divisor (GCD) of 69 and 111
The greatest common divisor (GCD) of 69 and 111 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 69 and 111?
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 | 69 ÷ 111 = 0 remainder 69 |
| 2 | 111 ÷ 69 = 1 remainder 42 |
| 3 | 69 ÷ 42 = 1 remainder 27 |
| 4 | 42 ÷ 27 = 1 remainder 15 |
| 5 | 27 ÷ 15 = 1 remainder 12 |
| 6 | 15 ÷ 12 = 1 remainder 3 |
| 7 | 12 ÷ 3 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 175 and 91 | 7 |
| 75 and 155 | 5 |
| 100 and 81 | 1 |
| 116 and 148 | 4 |
| 122 and 172 | 2 |