explain why 9997 is not a prime using euclidean algorithm
To understand why 9997 is not a prime using the Euclidean algorithm, let's first understand what the Euclidean algorithm is.
The Euclidean algorithm is an efficient method for finding the greatest common divisor (GCD) of two numbers. It is based on the principle that the GCD of two numbers is the same as the GCD of their difference and the smaller number.
Now, to determine whether 9997 is prime or not using the Euclidean algorithm, we need to find its GCD with another number. Let's choose a number that is smaller than 9997, say 17.
To calculate the GCD of 9997 and 17 using the Euclidean algorithm, we perform the following steps:
1. Divide 9997 by 17:
9997 ÷ 17 = 588 remainder 1
2. Divide 17 by the remainder (1):
17 ÷ 1 = 17 remainder 0
Since we have obtained a remainder of 0, we conclude that the GCD of 9997 and 17 is 1. This means that 9997 and 17 have no common factors other than 1.
In general, if a number is prime, its only divisors are 1 and itself. However, since the GCD of 9997 and 17 is 1, it indicates that 9997 has no common factors with 17, making it a possible prime number.
Therefore, based on the Euclidean algorithm, we cannot definitively determine whether 9997 is prime or not. Prime numbers have no divisors other than 1 and themselves. To confirm if 9997 is prime, we may need to use other methods like trial division or primality tests.