Number Classification: Identify the Type of Number 11

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

If a number has a factor larger than its square root, it must also have a smaller factor. For 11, $ \sqrt{11} \approx 3.3 $, so checking 2 and 3 is enough!

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

Prime numbers have exactly 2 factors (1 and themselves). Composite numbers have 3 or more factors. The number 1 is neither prime nor composite.

Q: How can I quickly check if 11 is divisible by 2?

Numbers divisible by 2 are even numbers (end in 0, 2, 4, 6, 8). Since 11 ends in 1, it's odd and not divisible by 2.

Q: What if I find that a number has other factors?

Then it's a composite number! For example, 15 has factors 1, 3, 5, and 15, so it's composite, not prime.

Prime Classification with Divisibility Testing

What type of number is the number n shown below?

$n=11$

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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

00:07 Therefore if the number is divisible by another factor it is not prime

00:11 The number has no other factors, therefore it is prime

00:16 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

What type of number is the number n shown below?

$n=11$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Verify if the number $n = 11$ is divisible by any numbers other than 1 and itself.
Step 2: Based on the result of the divisibility test, determine the type of number.

Now, let's work through each step:

Step 1: Check divisibility.
The number 11 is greater than 2, so we check divisibility by smaller primes up to $\sqrt{11} \approx 3.3$ . The prime numbers less than or equal to 3 are 2 and 3.

- **Divisibility by 2:** 11 is an odd number, hence not divisible by 2.
- **Divisibility by 3:** Sum of digits of 11 is $1 + 1 = 2$ , which is not divisible by 3.

Step 2: Conclusion based on divisibility.
Since 11 is not divisible by any other numbers except 1 and 11 itself, it does not have any divisors other than 1 and itself. Therefore, 11 is a prime number.

Hence, the solution to the problem is $n = 11$ is Prime.

Final Answer

Prime

Key Points to Remember

Essential concepts to master this topic

Prime Definition: Numbers with exactly two factors: 1 and itself
Technique: Test divisibility up to $\sqrt{n}$ ; for 11 check 2 and 3
Verification: 11 ÷ 2 = 5.5 and 11 ÷ 3 = 3.67 (both non-integers) ✓

Common Mistakes

Avoid these frequent errors

Assuming all odd numbers are prime
Don't think odd = prime automatically like saying 9 is prime! Many odd numbers like 9, 15, 21 have divisors besides 1 and themselves. Always test divisibility by checking if any smaller primes divide evenly into your number.

Practice Quiz

Test your knowledge with interactive questions

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

$$ n=10 $$

FAQ

Everything you need to know about this question

Why do I only need to check primes up to the square root?

If a number has a factor larger than its square root, it must also have a smaller factor. For 11, $\sqrt{11} \approx 3.3$ , so checking 2 and 3 is enough!

What's the difference between prime and composite numbers?

Prime numbers have exactly 2 factors (1 and themselves). Composite numbers have 3 or more factors. The number 1 is neither prime nor composite.

How can I quickly check if 11 is divisible by 2?

Numbers divisible by 2 are even numbers (end in 0, 2, 4, 6, 8). Since 11 ends in 1, it's odd and not divisible by 2.

Is there a trick for checking divisibility by 3?

Yes! Add up all the digits. If the sum is divisible by 3, the whole number is too. For 11: $1 + 1 = 2$ , and 2 ÷ 3 doesn't work, so 11 isn't divisible by 3.

What if I find that a number has other factors?

Then it's a composite number! For example, 15 has factors 1, 3, 5, and 15, so it's composite, not prime.

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