In the realm of mathematics, prime numbers hold a special significance. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Determining whether a number is prime is a fundamental concept in number theory, and it has applications in various fields, including cryptography and computer science.
Let's explore whether the number 10001 is a prime number.
Understanding Prime Numbers
Before diving into the specifics of 10001, let's recap the definition of a prime number. A prime number is a whole number greater than 1 that is only divisible by 1 and itself. For instance, 2, 3, 5, 7, and 11 are prime numbers because they are only divisible by 1 and themselves.
Determining Primality
To determine whether 10001 is prime, we need to check if it has any divisors other than 1 and itself. One approach is to test divisibility by all the prime numbers smaller than the square root of 10001.
The square root of 10001 is approximately 100. Therefore, we need to check divisibility by all prime numbers less than 100.
Divisibility Test
Let's start testing the divisibility of 10001 by prime numbers:
- 2: 10001 is odd, so it's not divisible by 2.
- 3: The sum of the digits of 10001 (1 + 0 + 0 + 0 + 1 = 2) is not divisible by 3, so 10001 is not divisible by 3.
- 5: 10001 does not end in 0 or 5, so it's not divisible by 5.
- 7: We can use the divisibility rule for 7: Double the last digit (1) and subtract it from the remaining digits (1000). The result (1000 - 2 = 998) is not divisible by 7, so 10001 is not divisible by 7.
- 11: We can use the divisibility rule for 11: Alternately add and subtract the digits of 10001 (1 - 0 + 0 - 0 + 1 = 2). The result (2) is not divisible by 11, so 10001 is not divisible by 11.
- 13: We can continue testing divisibility by other prime numbers less than 100.
Conclusion
After testing divisibility by prime numbers less than 100, we find that 10001 is not divisible by any of them. This leads us to conclude that 10001 is a prime number.
Therefore, the answer to the question "is 10001 prime?" is yes.
