Prime Number Identification: Testing Numbers for Primality

Q: Do I need to test all numbers as divisors?

No! You only need to test divisors up to the square root of your number. For $ 43 $, you only test up to about 6 since √43 ≈ 6.6.

Q: Why is 1 not considered a prime number?

By definition, prime numbers must have exactly two different divisors. The number 1 only has one divisor (itself), so it doesn't qualify as prime.

Q: How can I be sure 43 is really prime?

Test all possible divisors: $ 43 ÷ 2 = 21.5 $ (not whole), $ 43 ÷ 3 = 14.33... $ (not whole), $ 43 ÷ 5 = 8.6 $ (not whole). Since none give whole numbers, 43 is prime!

Prime Number Testing with Divisibility Checks

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

00:03 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

00:29 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 of the provided numbers is a prime number, follow these steps:

Step 1: Understand the definition of a prime number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
Step 2: Analyze and test each choice for primality:

45: Check divisibility by 2 (not even), by 3 (4 + 5 = 9, divisible by 3). Therefore, 45 is not prime.
46: Check divisibility by 2 (46 is even). Therefore, 46 is not prime.
42: Check divisibility by 2 (42 is even). Therefore, 42 is not prime.
43: Check divisibility by 2 (not even), by 3 (4 + 3 = 7, 7 is not divisible by 3), by 5 (does not end in 0 or 5). No divisors are found for 43. Hence, it is prime.

Upon checking, we see that 43 is the only number that cannot be evenly divided by any number other than 1 and itself. Thus, it is a prime number.

Therefore, the solution is $43$ .

Final Answer

$43$

Key Points to Remember

Essential concepts to master this topic

Definition: Prime numbers have exactly two divisors: 1 and themselves
Technique: Test divisibility by 2, 3, 5 systematically like $43 ÷ 3 = 14.33...$
Check: If no divisors found except 1 and the number itself, it's prime ✓

Common Mistakes

Avoid these frequent errors

Testing divisibility incompletely or incorrectly
Don't just check if a number 'looks prime' or only test one divisor = wrong conclusions! For example, 45 might seem prime but 45 ÷ 3 = 15. Always systematically test divisibility by 2, 3, 5, and other primes 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

How do I quickly check if a number is divisible by 3?

Add up all the digits! If the sum is divisible by 3, then the original number is too. For example: $45$ → 4 + 5 = 9, and 9 ÷ 3 = 3, so 45 is divisible by 3.

Do I need to test all numbers as divisors?

No! You only need to test divisors up to the square root of your number. For $43$ , you only test up to about 6 since √43 ≈ 6.6.

What's the easiest way to check if a number is even?

Look at the last digit! If it ends in 0, 2, 4, 6, or 8, it's even (and therefore not prime unless it's 2). Numbers like 46 and 42 are immediately ruled out.

Why is 1 not considered a prime number?

By definition, prime numbers must have exactly two different divisors. The number 1 only has one divisor (itself), so it doesn't qualify as prime.

How can I be sure 43 is really prime?

Test all possible divisors: $43 ÷ 2 = 21.5$ (not whole), $43 ÷ 3 = 14.33...$ (not whole), $43 ÷ 5 = 8.6$ (not whole). Since none give whole numbers, 43 is prime!

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Prime Number Identification: Testing Numbers for Primality

Prime Number Testing with Divisibility Checks

❤️ 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

How do I quickly check if a number is divisible by 3?

Do I need to test all numbers as divisors?

What's the easiest way to check if a number is even?

Why is 1 not considered a prime number?

How can I be sure 43 is really prime?

🌟 Unlock Your Math Potential

More Questions

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