Is 517 a prime number?

Understand the Problem

The question is asking whether the number 517 is a prime number. A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. To determine if 517 is prime, we will check if it can be divided evenly by any number other than 1 and 517.

Answer

The number 517 is not a prime number.

Steps to Solve

Identify the range of numbers to check for divisibility
To check if 517 is a prime number, we only need to test for factors up to the square root of 517.
Calculating the square root:
$$ \sqrt{517} \approx 22.7 $$
So, we will check for divisibility by integers from 2 to 22.
Check divisibility by 2
517 is an odd number, so it is not divisible by 2.
Check divisibility by other prime numbers
We will check for divisibility of 517 by the prime numbers up to 22: 3, 5, 7, 11, 13, 17, and 19.
Using the divisibility rule for each:

3: The sum of digits is $5 + 1 + 7 = 13$. Since 13 is not divisible by 3, 517 is not divisible by 3.
5: 517 does not end in 0 or 5, thus it is not divisible by 5.
7: Perform the division:
$$ 517 \div 7 \approx 73.857 \quad (\text{not an integer, so 517 is not divisible by 7}) $$
11: Perform the division:
$$ 517 \div 11 \approx 47.909 \quad (\text{not an integer, so 517 is not divisible by 11}) $$
13: Perform the division:
$$ 517 \div 13 = 39.0 \quad (\text{exactly, so 517 is divisible by 13}) $$

Conclusion about 517
Since 517 has a divisor (13), it is not a prime number.

The number 517 is not a prime number.

More Information

517 can be expressed as the product of its factors:
$$ 517 = 13 \times 39 $$
This shows that it has divisors other than 1 and itself.

Tips

Assuming all odd numbers are prime. Always check for factors regardless of whether a number is odd or even.
Not checking all prime numbers up to the square root. It's crucial to check these to confirm primality.

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