Given a sequence of letters, we replace each letter with its position in the alphabet, and then we compute the product of all the numbers. For example, if the sequence was HAL, then we get the numbers 8, 1, 12 (because H, A, L are the eight, first, twelfth letters of the alphabet, respectively), and their product is 96. Find the smallest composite three-digit number that cannot be the result of computing this product for any sequence of letters.
+23245 Let's see:
100 = 2 x 2 x 25 or 4 x 5 x 5 so it works
101 is prime so it is excluded
102 = 2 x 3 x 17 so it works
103 is prime so it is excluded
104 = 2 x 2 x 26 or 2 x 4 x 13 so it works
105 = 3 x 5 x 7 so it works
106 = 2 x 53 ============ this can't be encoded; so the answer is 106
