In the above cryptogram is divided by to obtain a recurring decimal which has repeating period of four digits . Furthermore, .
Compute .
Details And Assumptions:
represent distinct digits.
and are three digit numbers.
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The equation can be written as H U L K ⋅ D A D = B O B ⋅ 9 9 9 9 . We rule out D = 9 because that would give H U L K and B O B equal last digits. The fact that the quotient is less than 1 shows 1 ≤ B < D . Thus D ≥ 2 and B ≤ 7 .
Since gcd ( D A D , B O B ) = 1 , these have no common factors. Therefore 9 9 9 9 = n ⋅ D A D and H U L K = n ⋅ B O B , for a certain factor n ∣ 9 9 9 9 . From the numbers of digits of (\DAD) it is clear that 4 ≤ n < 5 0 .
Since 9999 is divisible by 101, D A D = 1 0 1 ⋅ D . This implies that A = 0 and n ⋅ D = 9 9 . This means that D = 3 , n = 3 3 : H U L K = 3 3 ⋅ B O B . Since B < D , B must be equal to 1 or 2. But B = 1 would make K = 3 = D , which is impossible. Therefore B = 2 .
For O we naturally rule out 0 , 2 , 3 . Also, since D ∣ B O B we have B O B ≡ 2 B + O ≡ 0 mod 3. Therefore O = 5 , 8 . This leaves five possibilities.
Only the second option has no overlap of digit values. Thus the solution of the cryptogram is 3 0 3 2 4 2 = 0 . 7 9 8 6 7 9 8 6 7 9 6 8 … The solution of the question is 2 + 4 + 3 + 0 + 7 + 9 + 8 + 6 = 3 9 .