Number Classification: Is 17 Prime or Composite?

Q: Why do I only need to test up to the square root?

If 17 had a factor larger than $ \sqrt{17} \approx 4.1 $, it would need a corresponding smaller factor to multiply and give 17. Since we already checked all smaller factors, we're done!

Q: What's the difference between prime and composite?

Prime numbers have exactly 2 factors (1 and themselves). Composite numbers have 3 or more factors. The number 1 is special - it's neither prime nor composite!

Q: Why isn't 1 considered prime?

By definition, prime numbers must have exactly two distinct factors. Since 1 only has one factor (itself), it doesn't meet the prime number criteria.

Prime Testing with Single Integer

Is the number equal to $n$ prime or composite?

$n=17$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the number composite or prime?

00:03 A prime number is only divisible by itself and 1

00:08 Therefore, if the number is divisible by another factor, it is not prime

00:16 The number has no other factors, therefore it is prime

00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Is the number equal to $n$ prime or composite?

$n=17$

Step-by-step solution

To determine whether $n = 17$ is a prime number, we will check if it has any divisors other than 1 and itself. A prime number has no divisors other than these two.

Step 1: Calculate $\sqrt{17}$ . Since $\sqrt{17} \approx 4.123$ , we need to test divisibility by integers 2, 3, and 4.
Step 2: Check divisibility by 2. Since 17 is odd, it is not divisible by 2.
Step 3: Check divisibility by 3. The sum of the digits of 17 is $1 + 7 = 8$ , which is not divisible by 3.
Step 4: Check divisibility by 4. Half of 17 is not a whole number, thus it is not divisible by 4.

Since 17 is not divisible by any number other than 1 and itself, it satisfies the condition of being a prime number.

Therefore, the solution to the problem is Prime.

Final Answer

Prime

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two factors: 1 and themselves
Testing Method: Check divisibility up to $\sqrt{17} \approx 4.1$ , test 2, 3, 4
Verification: 17 ÷ 2, 17 ÷ 3, 17 ÷ 4 all give remainders ✓

Common Mistakes

Avoid these frequent errors

Testing divisibility by all numbers up to 17
Don't check every number from 2 to 16 = wasted time and confusion! You only need factors, and if a number has a factor larger than its square root, it must also have one smaller. Always test only up to the square root.

Practice Quiz

Test your knowledge with interactive questions

Which of the numbers is a prime number?

FAQ

Everything you need to know about this question

Why do I only need to test up to the square root?

If 17 had a factor larger than $\sqrt{17} \approx 4.1$ , it would need a corresponding smaller factor to multiply and give 17. Since we already checked all smaller factors, we're done!

What's the difference between prime and composite?

Prime numbers have exactly 2 factors (1 and themselves). Composite numbers have 3 or more factors. The number 1 is special - it's neither prime nor composite!

Is there a faster way to check if 17 is prime?

For small numbers like 17, the division test is fastest. For larger numbers, you can use divisibility rules - like checking if the sum of digits is divisible by 3 for the number 3.

Why isn't 1 considered prime?

By definition, prime numbers must have exactly two distinct factors. Since 1 only has one factor (itself), it doesn't meet the prime number criteria.

What if I made an error in my division?

Double-check by using a calculator or trying the division again. Remember: if there's any remainder when dividing, then that number is not a factor of 17.

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