A3. How many two-digit numbers have a predecessor and a successor that are a prime number and a perfect square, not necessarily in that order?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
A3. The perfect squares that are the predecessor or successor of a two-digit number are exactly $9,16,25,36,49,64,81$ and 100. The two-digit numbers whose predecessor or successor is a perfect square are thus only the numbers $10,15,17,24,26,35,37,48,50,63,65,80,82$ and 99. However, a two-digit number whose predecessor or successor is a prime number must be even, so we are left with the numbers $10,24,26,48,50,80$ and 82. Of these, only the numbers $10,24,48,80$ and 82 have a prime number as their predecessor or successor. All these numbers indeed meet the condition. The correct answer is therefore 5.