Number Theory Practice: Identifying Prime Numbers

Prime Number Identification 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
00:07 Therefore if the number is divisible by another factor it is not 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
1

Understand the problem

Which of the numbers is a prime number?

2

Step-by-step solution

To determine which number is a prime number, we will check each choice for primality based on whether it has divisors other than 1 and itself:

  • Check 16 16 : This number is even and divisible by 2, hence not prime.
  • Check 17 17 : Testing divisibility by integers up to the square root of 17 (roughly 4.1), we find no divisors other than 1 and 17 17 itself. Therefore, 17 17 is prime.
  • Check 18 18 : This number is even and divisible by 2, hence not prime.
  • Check 14 14 : This number is also even and divisible by 2, hence not prime.

Given these observations, the only prime number among the choices is 17 17 .

3

Final Answer

17 17

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 n \sqrt{n} , for 17 check up to 4
  • Check: 17 ÷ 2, 3, 4 all have remainders, so 17 is prime ✓

Common Mistakes

Avoid these frequent errors
  • Confusing odd numbers with prime numbers
    Don't assume all odd numbers are prime = wrong classification! Numbers like 9, 15, 21 are odd but not prime because they have other divisors. Always test for divisors beyond 1 and the number itself.

Practice Quiz

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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?

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If a number has a factor larger than its square root, it must also have a factor smaller than the square root! For 17, 174.1 \sqrt{17} \approx 4.1 , so checking 2, 3, and 4 is enough.

Is 1 considered a prime number?

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No, 1 is not prime! Prime numbers must have exactly two factors: 1 and themselves. Since 1 only has one factor (itself), it doesn't qualify as prime.

How can I quickly tell if a number is NOT prime?

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Look for quick tests: Even numbers (except 2) are not prime, numbers ending in 5 (except 5) are not prime, and use divisibility rules for 3, 7, 11, etc.

What's the difference between 16 and 17?

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16 = 2 × 8 = 4 × 4, so it has many factors (1, 2, 4, 8, 16). But 17 cannot be divided evenly by any number except 1 and 17, making it prime.

Are there any patterns to help find prime numbers?

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Most primes (except 2 and 3) are one number away from multiples of 6, like 6±1. For example: 17 = 18-1, and 18 is a multiple of 6!

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