Find the Prime: Complete the Number Pattern in _3

To solve this problem, we aim to find a two-digit prime number of the form $\square 3$ using one of the given possible digits: 3, 4, 6, or 9.

Step 1: Form two-digit numbers.

• Using the digit 3: The number becomes $33$.
• Using the digit 4: The number becomes $43$.
• Using the digit 6: The number becomes $63$.
• Using the digit 9: The number becomes $93$.

Step 2: Check the primality of each two-digit number.

• $33$: Divisible by 3 (since $3 + 3 = 6$, and 6 is divisible by 3), so 33 is not a prime number.
• $43$: Check divisibility by prime numbers less than 7 (specifically, 2, 3, 5):
• Not divisible by 2 (because it is odd).
• Not divisible by 3 (since $4 + 3 = 7$; 7 is not divisible by 3).
• Not divisible by 5 (does not end with 0 or 5).
• Since none of these smaller primes divide 43, $43$ is prime.
• $63$: Divisible by 3 (since $6 + 3 = 9$, and 9 is divisible by 3), so 63 is not a prime number.
• $93$: Divisible by 3 (since $9 + 3 = 12$, and 12 is divisible by 3), so 93 is not a prime number.

Step 3: Conclusion

The only prime number formed is $43$.

Therefore, the number that fits in the blank to form a prime number is $4$.