Is 517 prime?

Steps to Solve

1. Identify the range of divisors to check

To determine if 517 is prime, we only need to check for divisors up to the square root of 517. The square root of 517 is approximately $22.7$, so we will check for divisibility by prime numbers up to $22$.

2. List the prime numbers to check

The prime numbers up to $22$ are: $2, 3, 5, 7, 11, 13, 17, 19$.

3. Check divisibility by 2

Since $517$ is odd, it is not divisible by $2$.

4. Check divisibility by 3

To check if $517$ is divisible by $3$, we sum the digits: $5 + 1 + 7 = 13$. Since $13$ is not divisible by $3$, $517$ is not divisible by $3$.

5. Check divisibility by 5

$517$ does not end in $0$ or $5$, so it is not divisible by $5$.

6. Check divisibility by 7

Now we divide $517$ by $7$: $$ 517 \div 7 = 73.8571 $$ Since this is not an integer, $517$ is not divisible by $7$.

7. Check divisibility by 11

Next, divide $517$ by $11$: $$ 517 \div 11 = 47.9091 $$ This also is not an integer, meaning it is not divisible by $11$.

8. Check divisibility by 13

Now we check $13$: $$ 517 \div 13 = 39.7692 $$ Again, this is not an integer, so not divisible by $13$.

9. Check divisibility by 17

Next, we check $17$: $$ 517 \div 17 = 30.4118 $$ Not an integer, so not divisible by $17$.

10. Check divisibility by 19

Finally, check $19$: $$ 517 \div 19 = 27.2105 $$ This is not an integer and means $517$ is not divisible by $19$.

11. Conclusion on primality

After testing all prime numbers up to $22$, we find that $517$ is not divisible by any of them. Thus, it cannot be divided by any number other than $1$ and itself, confirming that it is a prime number.