Mistakes do give rise to problems

B O B D A D = 0. H U L K H U L K H U L K \large \dfrac{BOB}{DAD} = 0.HULKHULKHULK\dots

In the above cryptogram B O B BOB is divided by D A D DAD to obtain a recurring decimal 0. H U L K H U L K H U L K 0.HULKHULKHULK\dots which has repeating period of four digits ( H U L K ) (HULK) . Furthermore, gcd ( B O B , D A D ) = 1 \gcd(BOB, DAD) = 1 .

Compute B + O + D + A + H + U + L + K B+O+D+A+H+U+L+K .

Details And Assumptions:

  • B , O , D , A , H , U , L , K B,O,D,A,H,U,L,K represent distinct digits.

  • B O B BOB and D A D DAD are three digit numbers.

Recently I was attempting Nihar's Problem and misread 1 statement of his question. This alternate problem is shown here since I'm a crappy person and don't want my efforts to be going down the drain.


The answer is 39.

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2 solutions

Arjen Vreugdenhil
Oct 21, 2015

The equation can be written as H U L K D A D = B O B 9999. HULK\cdot DAD = BOB\cdot 9999. We rule out D = 9 D = 9 because that would give H U L K HULK and B O B BOB equal last digits. The fact that the quotient is less than 1 shows 1 B < D 1\leq B < D . Thus D 2 D \geq 2 and B 7 B \leq 7 .

Since gcd ( D A D , B O B ) = 1 \text{gcd}(DAD,BOB) = 1 , these have no common factors. Therefore 9999 = n D A D 9999 = n\cdot DAD and H U L K = n B O B HULK = n\cdot BOB , for a certain factor n 9999 n | 9999 . From the numbers of digits of (\DAD) it is clear that 4 n < 50 4 \leq n < 50 .

Since 9999 is divisible by 101, D A D = 101 D DAD = 101\cdot D . This implies that A = 0 A = 0 and n D = 99 n\cdot D = 99 . This means that D = 3 , n = 33 D = 3, n = 33 : H U L K = 33 B O B . HULK = 33\cdot BOB. Since B < D B < D , B B must be equal to 1 or 2. But B = 1 B = 1 would make K = 3 = D K = 3 = D , which is impossible. Therefore B = 2 B = 2 .

For O O we naturally rule out 0 , 2 , 3 0, 2, 3 . Also, since D ∤ B O B D \not| BOB we have B O B 2 B + O ≢ 0 BOB \equiv 2B + O \not\equiv 0 mod 3. Therefore O 5 , 8 O \not= 5, 8 . This leaves five possibilities.

1
2
3
4
5
BOB = 212  HULK = 6996
BOB = 242  HULK = 7986 *
BOB = 262  HULK = 8646
BOB = 272  HULK = 8976
BOB = 292  HULK = 9636

Only the second option has no overlap of digit values. Thus the solution of the cryptogram is 242 303 = 0.798679867968 \frac{242}{303} = 0.798679867968\dots The solution of the question is 2 + 4 + 3 + 0 + 7 + 9 + 8 + 6 = 39 2 + 4 + 3 + 0 + 7 + 9 + 8 + 6 = \boxed{39} .

Ayy, it's the same as my problem!

Amazing Cryptogram #2

fingerguns

Boi (보이) - 3 years, 8 months ago
Vinod Kumar
Aug 14, 2020

(11^3)*6/9999=242/303

=0.79867986.....

Gives

Answer=39

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