Greatest Common Divisor (GCD) of 102 and 165
The greatest common divisor (GCD) of 102 and 165 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 102 and 165?
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 | 102 ÷ 165 = 0 remainder 102 |
| 2 | 165 ÷ 102 = 1 remainder 63 |
| 3 | 102 ÷ 63 = 1 remainder 39 |
| 4 | 63 ÷ 39 = 1 remainder 24 |
| 5 | 39 ÷ 24 = 1 remainder 15 |
| 6 | 24 ÷ 15 = 1 remainder 9 |
| 7 | 15 ÷ 9 = 1 remainder 6 |
| 8 | 9 ÷ 6 = 1 remainder 3 |
| 9 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 33 and 113 | 1 |
| 69 and 188 | 1 |
| 122 and 189 | 1 |
| 164 and 118 | 2 |
| 171 and 98 | 1 |