Prime Number Identification: Testing for Prime Properties

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

Test if small numbers like 2, 3, 5, 7 divide evenly into your number. For example, $ 9 ÷ 3 = 3 $ with no remainder, so 9 isn't prime!

Q: Is 1 considered a prime number?

No! By definition, prime numbers must be greater than 1 and have exactly two divisors. The number 1 only has one divisor (itself), so it's not prime.

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

Prime numbers have exactly 2 divisors (1 and themselves). Composite numbers have more than 2 divisors. For example, $ 8 $ has divisors 1, 2, 4, 8 - that's composite!

Q: Do I need to check every possible divisor?

No! You only need to check divisors up to the square root of your number. For $ 11 $, since $ \sqrt{11} ≈ 3.3 $, just test 2 and 3.

Q: Why is 11 the only prime in this list?

Let's check: $ 4 = 2 \times 2 $, $ 8 = 2 \times 4 $, $ 9 = 3 \times 3 $ all have extra divisors. But 11 only divides by 1 and 11 - making it prime!

Prime Number Recognition with Divisor Testing

Which of the numbers is a prime number?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the prime numbers

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

00:19 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

Which of the numbers is a prime number?

Step-by-step solution

To solve this problem, we'll identify which of the given numbers is a prime number:

Step 1: Define a prime number as a positive integer greater than 1 that has no divisors other than 1 and itself.
Step 2: Examine each number and list its divisors.

Now, let's work through each step:

Step 1: Consider the numbers given: $9$ , $11$ , $8$ , and $4$ .

Step 2:

$9$ has divisors $1, 3, 9$ . Since it has more than two divisors, it is not a prime number.
$11$ has divisors $1, 11$ . Since it has exactly two divisors, it is a prime number.
$8$ has divisors $1, 2, 4, 8$ . Since it has more than two divisors, it is not a prime number.
$4$ has divisors $1, 2, 4$ . Since it has more than two divisors, it is not a prime number.

Therefore, the number that is a prime number is $11$ .

Final Answer

$11$

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Testing: Check $9 = 3 \times 3$ shows 9 has divisor 3
Verify: Count all divisors - only primes have exactly 2 ✓

Common Mistakes

Avoid these frequent errors

Assuming larger numbers are always prime
Don't think 9 is prime just because it's bigger than some primes = wrong classification! Large numbers often have hidden factors. Always systematically check for divisors by testing if smaller numbers divide evenly into your candidate.

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

How do I quickly check if a number is prime?

Test if small numbers like 2, 3, 5, 7 divide evenly into your number. For example, $9 ÷ 3 = 3$ with no remainder, so 9 isn't prime!

Is 1 considered a prime number?

No! By definition, prime numbers must be greater than 1 and have exactly two divisors. The number 1 only has one divisor (itself), so it's not prime.

What's the difference between prime and composite numbers?

Prime numbers have exactly 2 divisors (1 and themselves). Composite numbers have more than 2 divisors. For example, $8$ has divisors 1, 2, 4, 8 - that's composite!

Do I need to check every possible divisor?

No! You only need to check divisors up to the square root of your number. For $11$ , since $\sqrt{11} ≈ 3.3$ , just test 2 and 3.

Why is 11 the only prime in this list?

Let's check: $4 = 2 \times 2$ , $8 = 2 \times 4$ , $9 = 3 \times 3$ all have extra divisors. But 11 only divides by 1 and 11 - making it prime!

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