Mathematical Investigation: Spotting Prime Numbers

Q: Why isn't 21 a prime number if it's odd?

Being odd doesn't guarantee a number is prime! 21 = 3 × 7, so it has divisors other than 1 and 21. Prime numbers must have exactly two divisors.

Q: Do I need to check every possible divisor?

No! You only need to test divisors up to the square root of the number. For 23, since $ \sqrt{23} \approx 4.8 $, just check 2, 3, and 4.

Q: What's the fastest way to spot non-prime numbers?

Even numbers (except 2) are never prime since they're divisible by 2. Also check if the sum of digits is divisible by 3 - like 21 where 2+1=3!

Q: Is there a pattern to help me remember prime numbers?

The first few primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23... Notice that except for 2, all primes are odd, but not all odd numbers are prime!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the prime numbers

00:04 A prime number is only divisible by itself and 1

00:07 Therefore, we'll check if the number is divisible by another factor, not prime

00:37 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Which of the numbers is a prime number?

2

Step-by-step solution

To solve this problem, we'll verify whether each number is a prime number:

22: Divisible by 2. Not a prime number.
23: Divisible only by 1 and 23. It is a prime number.
21: Divisible by 3 and 7. Not a prime number.
24: Divisible by 2. Not a prime number.

Now, let's go through each step in detail:

Step 1: Check 22
22 is even, meaning it is divisible by 2. As it has divisors other than 1 and itself, 22 is not a prime number.

Step 2: Check 23
Begin by testing divisibility by 2. Since 23 is odd, it is not divisible by 2. Next, check divisibility by 3: $23 \div 3 \approx 7.67$ , not an integer. No divisors other than 1 and 23 are found, therefore 23 is a prime number.

Step 3: Check 21
21 is odd, so not divisible by 2. However, $21 \div 3 = 7$ , which is an integer. Thus, 21 is not prime because it is divisible by numbers other than 1 and itself.

Step 4: Check 24
24 is even and divisible by 2. Hence, 24 is not a prime number.

Therefore, the solution to the problem is $23$ .

3

Final Answer

$23$

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Method: Test divisibility by small primes like $21 \div 3 = 7$
Verification: Check all potential divisors up to the square root ✓

Common Mistakes

Avoid these frequent errors

Assuming odd numbers are always prime
Don't think all odd numbers like 21 are prime = wrong classification! Just being odd doesn't make a number prime since 21 = 3 × 7. Always test for divisors other than 1 and the number itself.

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 know if a number like 23 is really prime?

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Test if 23 is divisible by small prime numbers: 2, 3, 5, 7, etc. Since $23 \div 2 = 11.5$ and $23 \div 3 \approx 7.67$ aren't whole numbers, 23 is prime!

Why isn't 21 a prime number if it's odd?

+

Being odd doesn't guarantee a number is prime! 21 = 3 × 7, so it has divisors other than 1 and 21. Prime numbers must have exactly two divisors.

Do I need to check every possible divisor?

+

No! You only need to test divisors up to the square root of the number. For 23, since $\sqrt{23} \approx 4.8$ , just check 2, 3, and 4.

What's the fastest way to spot non-prime numbers?

+

Even numbers (except 2) are never prime since they're divisible by 2. Also check if the sum of digits is divisible by 3 - like 21 where 2+1=3!

Is there a pattern to help me remember prime numbers?

+

The first few primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23... Notice that except for 2, all primes are odd, but not all odd numbers are prime!

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Mathematical Investigation: Spotting Prime Numbers

Prime Number Identification with Divisibility Testing

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