Mathematical Properties: Prime Number Detection Exercise

Q: 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 smaller divisor. For example, if 35 ÷ 7 = 5, then we already found both divisors by checking up to $ \sqrt{35} \approx 5.9 $!

Q: Are all odd numbers prime?

No! Many odd numbers like 9, 15, 21, 25 are composite (not prime). Being odd just means it's not divisible by 2, but it could still be divisible by other numbers like 3, 5, or 7.

Q: Is 1 considered a prime number?

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

Q: What's the fastest way to check if 31 is prime?

Check if it ends in 0, 2, 4, 5, 6, 8 (not prime if yes)Add digits: 3 + 1 = 4, not divisible by 3Doesn't end in 0 or 5, so not divisible by 5Since we only need to check up to 5, and none work, 31 is prime!

Q: Why are 30, 32, and 34 definitely not prime?

All three are even numbers (end in 0, 2, 4), which means they're automatically divisible by 2. Since they have divisors other than 1 and themselves, they cannot be prime by definition.

Prime Numbers with Divisibility 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

00:04 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

00:30 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 determine which number is a prime number among the choices, we proceed with the following analysis:

First, let's define a prime number:
A prime number is a number greater than 1 that has no divisors other than 1 and itself.

Check $34$ : $34$ is even, hence divisible by 2. So, it is not a prime number.
Check $30$ : $30$ is also even and divisible by 2. Thus, it is not a prime number.
Check $32$ : $32$ is even, so it is divisible by 2. Therefore, it is not a prime number.
Check $31$ : Let's check for divisibility by numbers up to the square root of $31$ , which is approximately 5.57.

It is not divisible by 2 because it is not even.
It is not divisible by 3, as $31 \div 3$ does not result in an integer.
It is not divisible by 5, as it does not end in 0 or 5.

Since $31$ is not divisible by any prime number up to its square root, $31$ is a prime number.

Therefore, the solution to the problem is $31$ , which is a prime number.

Final Answer

$31$

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Technique: Check divisibility up to square root, like $\sqrt{31} \approx 5.57$
Check: Test $31 ÷ 2, 3, 5$ all give remainders ✓

Common Mistakes

Avoid these frequent errors

Only checking if a number is even or odd
Don't just check if 31 is odd and call it prime = incomplete test! Odd numbers like 9, 15, 21 aren't prime because they have other divisors. Always test divisibility by all prime numbers up to the square root.

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 divisor larger than its square root, it must also have a corresponding smaller divisor. For example, if 35 ÷ 7 = 5, then we already found both divisors by checking up to $\sqrt{35} \approx 5.9$ !

Are all odd numbers prime?

No! Many odd numbers like 9, 15, 21, 25 are composite (not prime). Being odd just means it's not divisible by 2, but it could still be divisible by other numbers like 3, 5, or 7.

Is 1 considered a prime number?

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

What's the fastest way to check if 31 is prime?

Check if it ends in 0, 2, 4, 5, 6, 8 (not prime if yes)
Add digits: 3 + 1 = 4, not divisible by 3
Doesn't end in 0 or 5, so not divisible by 5
Since we only need to check up to 5, and none work, 31 is prime!

Why are 30, 32, and 34 definitely not prime?

All three are even numbers (end in 0, 2, 4), which means they're automatically divisible by 2. Since they have divisors other than 1 and themselves, they cannot be prime by definition.

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Mathematical Properties: Prime Number Detection Exercise

Prime Numbers with Divisibility Testing

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

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

Are all odd numbers prime?

Is 1 considered a prime number?

What's the fastest way to check if 31 is prime?

Why are 30, 32, and 34 definitely not prime?

🌟 Unlock Your Math Potential

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