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.
Answer for screen readers
The number 517 is not a prime number.
Steps to Solve
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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.
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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:
$$ 517 \div 13 = 39.0 \quad (\text{exactly, so 517 is divisible by 13}) $$
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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.
