Greatest Common Divisor (GCD) of 67 and 165
The greatest common divisor (GCD) of 67 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 67 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 | 67 ÷ 165 = 0 remainder 67 |
| 2 | 165 ÷ 67 = 2 remainder 31 |
| 3 | 67 ÷ 31 = 2 remainder 5 |
| 4 | 31 ÷ 5 = 6 remainder 1 |
| 5 | 5 ÷ 1 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 133 and 112 | 7 |
| 170 and 138 | 2 |
| 93 and 134 | 1 |
| 128 and 160 | 32 |
| 64 and 145 | 1 |