Greatest Common Divisor (GCD) of 53 and 137
The greatest common divisor (GCD) of 53 and 137 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 53 and 137?
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 | 53 ÷ 137 = 0 remainder 53 |
| 2 | 137 ÷ 53 = 2 remainder 31 |
| 3 | 53 ÷ 31 = 1 remainder 22 |
| 4 | 31 ÷ 22 = 1 remainder 9 |
| 5 | 22 ÷ 9 = 2 remainder 4 |
| 6 | 9 ÷ 4 = 2 remainder 1 |
| 7 | 4 ÷ 1 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 153 and 102 | 51 |
| 35 and 100 | 5 |
| 173 and 45 | 1 |
| 51 and 183 | 3 |
| 30 and 96 | 6 |