Is 10 Prime or Composite? Number Classification Problem

Q: Why isn't 10 considered prime if it's such a common number?

Being common doesn't make a number prime! Prime numbers are defined by having exactly two divisors. Since 10 has divisors 1, 2, 5, and 10, it has more than two divisors and is composite.

Q: How do I quickly check if a number is composite?

Start with the smallest prime numbers: 2, 3, 5, 7. If your number divides evenly by any of these, it's composite. For 10: $ 10 \div 2 = 5 $ exactly, so it's composite!

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

Prime: Exactly 2 divisors (1 and itself)Composite: More than 2 divisorsSpecial case: 1 is neither prime nor composite

Q: Do I need to check all possible divisors?

No! You only need to check up to the square root of the number. For 10, you only need to check 2 and 3 since $ \sqrt{10} \approx 3.16 $.

Q: Can a composite number have exactly 3 divisors?

Yes! For example, 9 has divisors 1, 3, and 9. Any composite number can have anywhere from 3 to many divisors, while prime numbers always have exactly 2.

Prime Classification with Divisor Analysis

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

$n=10$

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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

00:12 The number has other factors, meaning it's composite

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

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

$n=10$

Step-by-step solution

A number is classified as prime if it has exactly two distinct positive divisors: 1 and itself. Conversely, a number is composite if it has more than two divisors.

Given the number $n = 10$ , we need to determine whether it is prime or composite.

Let's test the divisibility of 10 by numbers other than 1 and 10:

Check divisibility by 2: Since 10 is an even number, it is divisible by 2. Specifically, $10 \div 2 = 5$ with no remainder.
Check divisibility by 3: $10 \div 3 \approx 3.333$ , which is not an integer, hence not divisible.
Check divisibility by 5: $10 \div 5 = 2$ with no remainder, showing divisibility by 5.

Since 10 is divisible by numbers other than 1 and itself (specifically 2 and 5), it is not prime. Therefore, the number 10 is composite.

In conclusion, the number 10 is a composite number.

Final Answer

Composite

Key Points to Remember

Essential concepts to master this topic

Definition: Prime has exactly two divisors: 1 and itself
Technique: Check divisibility by small numbers like $10 \div 2 = 5$
Check: If more than two divisors exist, the number is composite ✓

Common Mistakes

Avoid these frequent errors

Assuming all even numbers except 2 are prime
Don't think that 10 could be prime just because it's a nice round number = wrong classification! Even numbers (except 2) are always divisible by 2, making them composite by definition. Always check for divisibility by 2 first for even numbers.

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 isn't 10 considered prime if it's such a common number?

Being common doesn't make a number prime! Prime numbers are defined by having exactly two divisors. Since 10 has divisors 1, 2, 5, and 10, it has more than two divisors and is composite.

How do I quickly check if a number is composite?

Start with the smallest prime numbers: 2, 3, 5, 7. If your number divides evenly by any of these, it's composite. For 10: $10 \div 2 = 5$ exactly, so it's composite!

What's the difference between prime and composite again?

Prime: Exactly 2 divisors (1 and itself)
Composite: More than 2 divisors
Special case: 1 is neither prime nor composite

Do I need to check all possible divisors?

No! You only need to check up to the square root of the number. For 10, you only need to check 2 and 3 since $\sqrt{10} \approx 3.16$ .

Can a composite number have exactly 3 divisors?

Yes! For example, 9 has divisors 1, 3, and 9. Any composite number can have anywhere from 3 to many divisors, while prime numbers always have exactly 2.

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