Number Theory Practice: Identifying Prime Numbers from a Set

Q: How do I quickly check if 39 is divisible by 3?

Add the digits: 3 + 9 = 12. Since 12 is divisible by 3, the original number 39 is also divisible by 3. This is the divisibility rule for 3!

Q: What makes 37 different from the other numbers?

37 is odd (not divisible by 2), the sum of digits is 10 (not divisible by 3), and it doesn't end in 0 or 5 (not divisible by 5). No other small primes divide it either!

Q: Are there any patterns to help me recognize primes?

Yes! All primes except 2 are odd. Also, primes greater than 3 often end in 1, 3, 7, or 9. But remember: not all numbers with these endings are prime - you still need to test!

Q: What if I forget the divisibility rules?

You can always do direct division! For example, 39 ÷ 3 = 13 exactly, so 39 isn't prime. The rules just make checking faster.

Q: Is 1 considered a prime number?

No! By definition, prime numbers must have exactly two factors: 1 and themselves. Since 1 only has one factor (itself), it's not considered 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

00:09 Therefore, if the number is divisible by another factor, it is not prime

00:31 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 determine which of the given numbers is a prime number, follow these steps:

Step 1: Check the number 39.
39 is divisible by 3 (since 3 + 9 = 12, which is divisible by 3), so it has divisors other than 1 and itself, and thus 39 is not prime.
Step 2: Check the number 34.
34 is divisible by 2 (as it's an even number), so it has divisors other than 1 and itself, and thus 34 is not prime.
Step 3: Check the number 42.
42 is divisible by 2 (as it's an even number), so it has divisors other than 1 and itself, and thus 42 is not prime.
Step 4: Check the number 37.
For a number to be prime, it should not be divisible by any numbers other than 1 and itself. We check divisibility of 37 by numbers up to its square root (~6.08), which are 2, 3, and 5.

37 is not divisible by 2 (as it is odd).
37 is not divisible by 3 (as 3 + 7 = 10, which is not divisible by 3).
37 is not divisible by 5 (as it does not end in 0 or 5).

Since 37 is not divisible by any integer other than 1 and 37 itself, it is a prime number.

Therefore, the solution to the problem is $37$ .

3

Final Answer

$37$

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two factors: 1 and themselves
Technique: Test divisibility up to square root (√37 ≈ 6)
Check: 37 ÷ 2, 3, 5 leaves remainders, so 37 is prime ✓

Common Mistakes

Avoid these frequent errors

Assuming odd numbers are always prime
Don't think all odd numbers like 39 are prime = wrong classification! 39 = 3 × 13, so it has factors beyond 1 and itself. Always test divisibility by checking if the number has other factors besides 1 and 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

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 $\sqrt{37} ≈ 6$ is sufficient!

How do I quickly check if 39 is divisible by 3?

+

Add the digits: 3 + 9 = 12. Since 12 is divisible by 3, the original number 39 is also divisible by 3. This is the divisibility rule for 3!

What makes 37 different from the other numbers?

+

37 is odd (not divisible by 2), the sum of digits is 10 (not divisible by 3), and it doesn't end in 0 or 5 (not divisible by 5). No other small primes divide it either!

Are there any patterns to help me recognize primes?

+

Yes! All primes except 2 are odd. Also, primes greater than 3 often end in 1, 3, 7, or 9. But remember: not all numbers with these endings are prime - you still need to test!

What if I forget the divisibility rules?

+

You can always do direct division! For example, 39 ÷ 3 = 13 exactly, so 39 isn't prime. The rules just make checking faster.

Is 1 considered a prime number?

+

No! By definition, prime numbers must have exactly two factors: 1 and themselves. Since 1 only has one factor (itself), it's not considered prime.

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Number Theory Practice: Identifying Prime Numbers from a Set

Prime Number Recognition with Divisibility Tests

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