Prime Number Identification: Testing Numbers for Prime Properties

Q: How do I quickly check if 21 is divisible by 3?

Use the divisibility rule for 3: add all digits together. Since 2 + 1 = 3, and 3 is divisible by 3, then 21 is also divisible by 3.

Q: Is 1 considered a prime number?

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

Q: What's the fastest way to check if 19 is prime?

Check divisibility by all primes up to $ \sqrt{19} \approx 4.4 $. Test: 19 ÷ 2 = 9.5 (not whole), 19 ÷ 3 = 6.33... (not whole). Since no prime ≤ 4 divides 19, it's prime!

Q: Why is 16 definitely not prime?

Any even number greater than 2 cannot be prime because it's automatically divisible by 2. Since 16 = 2 × 8, it has divisors other than 1 and itself.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:03 First, let's pick out the prime numbers.

00:06 Remember, a prime number divides evenly only by itself and one.

00:11 So, if a number can be divided by any other factors, it's not a prime. Keep looking for those unique ones!

00:36 And there you have it! That's how we solve this problem.

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 determine which of the given numbers is a prime number, we will evaluate each one to check if it has any divisors other than 1 or itself.

Here are the steps:

Step 1: Examine number 18:
- 18 is divisible by 2 (as it is even) and also by 3 (since 1 + 8 = 9, which is divisible by 3). Therefore, 18 is not a prime number.
Step 2: Examine number 21:
- 21 is divisible by 3 (since 2 + 1 = 3, which is divisible by 3) and by 7 (as $21 \div 7 = 3$ ). Thus, 21 is not a prime number.
Step 3: Examine number 16:
- 16 is divisible by 2 (multiple times: 16, 8, 4, 2) and hence is not a prime number.
Step 4: Examine number 19:
- Check divisibility by 2: 19 is odd, not divisible by 2.
- Check divisibility by 3: 1 + 9 = 10, not divisible by 3.
- Check divisibility by 5: Does not end in 0 or 5.
- Check divisibility by any prime numbers less than the square root of 19: None divides 19 evenly.
As no number other than 1 and itself divides 19, it is a prime number.

Therefore, the solution is that $19$ is the prime number among the choices given.

3

Final Answer

$19$

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Testing: Check divisibility by primes up to square root: $\sqrt{19} \approx 4.4$
Verification: Test 19 ÷ 2, 19 ÷ 3: no whole number results ✓

Common Mistakes

Avoid these frequent errors

Only checking divisibility by 2 and 3
Don't stop after testing just 2 and 3 = missing other prime divisors! You might incorrectly identify composite numbers as prime. Always test all prime numbers up to the square root of 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 up to the square root?

+

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

How do I quickly check if 21 is divisible by 3?

+

Use the divisibility rule for 3: add all digits together. Since 2 + 1 = 3, and 3 is divisible by 3, then 21 is also divisible by 3.

Is 1 considered a prime number?

+

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

What's the fastest way to check if 19 is prime?

+

Check divisibility by all primes up to $\sqrt{19} \approx 4.4$ . Test: 19 ÷ 2 = 9.5 (not whole), 19 ÷ 3 = 6.33... (not whole). Since no prime ≤ 4 divides 19, it's prime!

Why is 16 definitely not prime?

+

Any even number greater than 2 cannot be prime because it's automatically divisible by 2. Since 16 = 2 × 8, it has divisors other than 1 and itself.

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Prime Number Identification: Testing Numbers for Prime Properties

Prime Number Identification with Divisibility Testing

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