Is 23 Prime or Composite? Number Classification Challenge

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

If a number has a factor larger than its square root, it must also have a corresponding factor smaller than the square root. So checking up to $ \sqrt{n} $ covers all possibilities!

Q: Do I need to check all numbers or just prime numbers?

You only need to check prime numbers up to the square root! If a composite number divides your number, then its prime factors would divide it too, so you'd catch it when testing those primes.

Q: What if the square root is not a whole number?

Round down to the nearest integer! For $ \sqrt{23} \approx 4.8 $, you check primes up to 4, which are just 2 and 3.

Q: Is 1 considered a prime number?

No! By definition, prime numbers must have exactly two distinct divisors. Since 1 only has one divisor (itself), it's neither prime nor composite.

Prime Number Testing with Square Root Method

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

$n=23$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the number prime or composite?

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 no other factors, meaning it is prime

00:17 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?

$n=23$

Step-by-step solution

To determine whether $n = 23$ is a prime number, we will test its divisibility:

Step 1: Calculate $\sqrt{23}$ . The approximate value is 4.795, and thus we consider prime numbers up to the integer part, which is 4.
Step 2: Check if 23 is divisible by any prime numbers less than or equal to 4. These primes are 2 and 3.

Step 3: Test divisibility:
- 23 is not divisible by 2, as it is odd.
- 23 is not divisible by 3, since $23 \div 3 \approx 7.67$ , which is not an integer.

Since 23 is not divisible by any prime number less than or equal to its square root, it only has divisors of 1 and 23. Hence, 23 is a prime number.

Therefore, the solution to the problem is that $n = 23$ is prime.

Final Answer

Prime

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Test Method: Check divisibility by primes up to $\sqrt{23} \approx 4.8$
Verification: 23 ÷ 2 = 11.5 and 23 ÷ 3 = 7.67, both non-integers ✓

Common Mistakes

Avoid these frequent errors

Testing divisibility by all numbers up to the given number
Don't check if 23 is divisible by every number from 2 to 22 = wasted time and effort! You only need to test numbers up to the square root. Always test divisibility only by prime numbers up to $\sqrt{n}$ for efficiency.

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 check up to the square root?

If a number has a factor larger than its square root, it must also have a corresponding factor smaller than the square root. So checking up to $\sqrt{n}$ covers all possibilities!

Do I need to check all numbers or just prime numbers?

You only need to check prime numbers up to the square root! If a composite number divides your number, then its prime factors would divide it too, so you'd catch it when testing those primes.

What if the square root is not a whole number?

Round down to the nearest integer! For $\sqrt{23} \approx 4.8$ , you check primes up to 4, which are just 2 and 3.

How do I quickly check if a number is divisible by 3?

Add up all the digits! If the sum is divisible by 3, then the original number is too. For 23: 2 + 3 = 5, and 5 ÷ 3 is not a whole number, so 23 isn't divisible by 3.

Is 1 considered a prime number?

No! By definition, prime numbers must have exactly two distinct divisors. Since 1 only has one divisor (itself), it's neither prime nor composite.

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