Greatest Common Divisor (GCD) of 165 and 103
The greatest common divisor (GCD) of 165 and 103 is 1.
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 165 and 103?
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 | 165 ÷ 103 = 1 remainder 62 |
| 2 | 103 ÷ 62 = 1 remainder 41 |
| 3 | 62 ÷ 41 = 1 remainder 21 |
| 4 | 41 ÷ 21 = 1 remainder 20 |
| 5 | 21 ÷ 20 = 1 remainder 1 |
| 6 | 20 ÷ 1 = 20 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 39 and 174 | 3 |
| 153 and 92 | 1 |
| 114 and 96 | 6 |
| 115 and 39 | 1 |
| 142 and 139 | 1 |