Greatest Common Divisor (GCD) of 157 and 91
The greatest common divisor (GCD) of 157 and 91 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 157 and 91?
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 | 157 ÷ 91 = 1 remainder 66 |
| 2 | 91 ÷ 66 = 1 remainder 25 |
| 3 | 66 ÷ 25 = 2 remainder 16 |
| 4 | 25 ÷ 16 = 1 remainder 9 |
| 5 | 16 ÷ 9 = 1 remainder 7 |
| 6 | 9 ÷ 7 = 1 remainder 2 |
| 7 | 7 ÷ 2 = 3 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 118 and 133 | 1 |
| 87 and 103 | 1 |
| 171 and 116 | 1 |
| 99 and 195 | 3 |
| 104 and 199 | 1 |