
Greatest Common Divisor (GCD) of 106 and 165
The greatest common divisor (GCD) of 106 and 165 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 106 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 | 106 ÷ 165 = 0 remainder 106 |
2 | 165 ÷ 106 = 1 remainder 59 |
3 | 106 ÷ 59 = 1 remainder 47 |
4 | 59 ÷ 47 = 1 remainder 12 |
5 | 47 ÷ 12 = 3 remainder 11 |
6 | 12 ÷ 11 = 1 remainder 1 |
7 | 11 ÷ 1 = 11 remainder 0 |
Examples of GCD Calculations
Numbers | GCD |
---|---|
182 and 178 | 2 |
144 and 30 | 6 |
64 and 54 | 2 |
161 and 101 | 1 |
166 and 167 | 1 |