Find the Missing Digit: Complete the Prime Number _7

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.