Find the Missing Digit: Complete the Prime Number _7

Q: Do I need to test every single digit from 0 to 9?

Yes! You need to test all possible digits to be thorough. Form 07, 17, 27, 37, 47, 57, 67, 77, 87, 97 and check each one. Some can be eliminated quickly (like even numbers), but testing all ensures you don't miss anything.

Q: Why isn't 57 prime when it ends in 7 like 37?

The last digit doesn't determine if a number is prime! 57 = 3 × 19, so it has factors other than 1 and itself. Always check divisibility by all primes up to the square root, not just the last digit.

Q: Do I need to memorize which numbers are prime?

No need to memorize! Learn the testing method instead. Check divisibility by 2, 3, 5, 7, 11... up to $ \sqrt{n} $. This works for any number and helps you understand why it's prime.

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

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

Prime Number Testing with Two-Digit Numbers

Fill in the blank for a prime number:

$\square7$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the blank for a prime number:

$\square7$

Step-by-step solution

To solve this problem, we'll conduct primality tests for each possible number formed by different digits in place of $\square$ in $\square7$ .

Step 1: Test if $37$ is prime.
Step 2: Test if $57$ is prime.
Step 3: Test if $87$ is prime.
Step 4: Test if $77$ is prime.

Let's detail these steps:

Step 1: Check $37$ .

$37$ is not divisible by any prime numbers up to its square root ( $\sqrt{37} \approx 6.08$ ), specifically 2, 3, 5. Therefore, $37$ is prime.

Step 2: Check $57$ .

$57$ is divisible by 3 ( $57 \div 3 = 19$ ). Thus, $57$ is not prime.

Step 3: Check $87$ .

$87$ is divisible by 3 ( $87 \div 3 = 29$ ). Hence, $87$ is not prime.

Step 4: Check $77$ .

$77$ is divisible by 7 ( $77 \div 7 = 11$ ). Consequently, $77$ is not prime.

Therefore, the number that completes $\square7$ as a prime number is $\boxed{3}$ , forming $37$ which is prime.

Final Answer

$3$

Key Points to Remember

Essential concepts to master this topic

Prime Definition: Number divisible only by 1 and itself
Testing Method: Check divisibility by primes up to $\sqrt{n}$ (37: test 2,3,5)
Verification: Only 37 has no divisors, making it prime ✓

Common Mistakes

Avoid these frequent errors

Not testing all possible digits systematically
Don't just guess random digits or only test obvious ones like 1 and 9 = missing the correct answer! Students often skip testing all single digits 0-9. Always check each digit 0-9 to form complete two-digit numbers ending in 7, then test each for primality.

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

Do I need to test every single digit from 0 to 9?

Yes! You need to test all possible digits to be thorough. Form 07, 17, 27, 37, 47, 57, 67, 77, 87, 97 and check each one. Some can be eliminated quickly (like even numbers), but testing all ensures you don't miss anything.

Why isn't 57 prime when it ends in 7 like 37?

The last digit doesn't determine if a number is prime! 57 = 3 × 19, so it has factors other than 1 and itself. Always check divisibility by all primes up to the square root, not just the last digit.

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 whole number is too. For example: 57 → 5+7 = 12, and 12÷3 = 4, so 57 is divisible by 3.

What if multiple digits make prime numbers?

Great question! In this case, only 3 works to make 37 prime. But if multiple answers were prime, you'd choose all of them. The question asks for the digit, suggesting there's only one correct answer.

Do I need to memorize which numbers are prime?

No need to memorize! Learn the testing method instead. Check divisibility by 2, 3, 5, 7, 11... up to $\sqrt{n}$ . This works for any number and helps you understand why it's prime.

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

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

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