Gọi số ban đầu có ba chữ số là ABC.

Theo đề bài, khi xóa đi chữ số 3 ở hàng trăm, ta được số mới là AB.

Ta có thể viết phương trình như sau:
ABC = 100A + 10B + C
AB = 10A + B

Theo đề bài, thương của số ban đầu chia cho số mới là 5 dư 12:
(100A + 10B + C) / (10A + B) = 5 dư 12

Ta có thể viết phương trình như sau:
100A + 10B + C = 5(10A + B) + 12
100A + 10B + C = 50A + 5B + 12

Simplifying the equation, we get:
50A + 5B + 12 = 100A + 10B + C

Rearranging the equation, we get:
50A - 100A + 5B - 10B = C - 12
-50A - 5B = C - 12

Since A, B, and C are digits, they must be integers between 0 and 9.

We can try different values for A and B to find the corresponding value of C that satisfies the equation.

Let's try A = 1 and B = 2:
-50(1) - 5(2) = C - 12
-50 - 10 = C - 12
-60 = C - 12
C = -60 + 12
C = -48

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 2 and B = 3:
-50(2) - 5(3) = C - 12
-100 - 15 = C - 12
-115 = C - 12
C = -115 + 12
C = -103

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 3 and B = 4:
-50(3) - 5(4) = C - 12
-150 - 20 = C - 12
-170 = C - 12
C = -170 + 12
C = -158

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 4 and B = 5:
-50(4) - 5(5) = C - 12
-200 - 25 = C - 12
-225 = C - 12
C = -225 + 12
C = -213

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 5 and B = 6:
-50(5) - 5(6) = C - 12
-250 - 30 = C - 12
-280 = C - 12
C = -280 + 12
C = -268

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 6 and B = 7:
-50(6) - 5(7) = C - 12
-300 - 35 = C - 12
-335 = C - 12
C = -335 + 12
C = -323

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 7 and B = 8:
-50(7) - 5(8) = C - 12
-350 - 40 = C - 12
-390 = C - 12
C = -390 + 12
C = -378

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 8 and B = 9:
-50(8) - 5(9) = C - 12
-400 - 45 = C - 12
-445 = C - 12
C = -445 + 12
C = -433

Since C is not a digit between 0 and 9, this solution does not work.

Let's try A = 9 and B = 0:
-50(9) - 5(0) = C - 12
-450 - 0 = C - 12
-450 = C - 12
C = -450 + 12
C = -438

Since C is not a digit between 0 and 9, this solution does not work.

After trying all possible values for A and B, we can conclude that there is no three-digit number that satisfies the given conditions.