What is the smallest integer greater than 237 for which each digit of the integer is a prime number?

252  

Step-by-step explanation:

Here's how I would argue.

Let the three digits be a, b, and c,

Then the number is abc.

Assume a = 2 and b= 3.  

Then c cannot be 8, 9, or 0, because they are not prime numbers.

b cannot equal 4 (not prime), so  

Assume that b = 5 (prime).

a cannot be 0 or 1 (not prime), but it can equal 2 (prime).

abc = 252

Thus, 252 is the smallest integer greater than 237 for which each digit is a prime.