EIGHT is not 8

Number Theory Level 3

E I G H T ÷ E \color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T\color{#333333}\div\color{#20A900}E leaves remainder I . \color{#3D99F6}I.
E I G H T ÷ I \color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T\color{#333333}\div\color{#3D99F6}I leaves remainder G . \color{#D61F06}G.
E I G H T ÷ G \color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T\color{#333333}\div\color{#D61F06}G leaves remainder H . \color{#E81990}H.
E I G H T ÷ H \color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T\color{#333333}\div\color{#E81990}H leaves remainder 0. 0.
E I G H T ÷ T \color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T\color{#333333}\div\color{#BA33D6}T leaves remainder 0. 0.

Every letter represents a distinct, single digit. There are some possible numbers for E I G H T ‾ \overline{\color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T} .

What is the sum of these possible numbers?

Here is another problem like this.


The answer is 181530.

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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2 solutions

Arjen Vreugdenhil
Dec 13, 2017 
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for E in range(1,10):
  for I in range(1,10):
    for G in range(1,10):
      for H in range(1,10):
        for T in range(1,10):
          if len(set([E,I,G,H,T])) == 5:
            n = (((E*10+I)*10+G)*10+H)*10+T
            r1 = n % E
            r2 = n % I
            r3 = n % G
            r4 = n % H
            r5 = n % T
            if (r1 == I) and (r2 == G) and (r3 == H) and (r4 == 0) and (r5 == 0):
                print(n)

Output: 

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87215
94315

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Because of the conditions on the remainders, it can be said that 0 < H < G < I < E ≤ 9 0<H<G<I<E \leq 9 . This will substantially reduce the search space. Further, all digits are non-zero and less than 10.

Thus, it could be encoded as 

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   for E=1:9
      for I=1:(E-1)
         for G=1:(I-1)
            for H=1:(G-1)
               for T=setdiff(1:9,[E I G H])
                  n = 10000*E+1000*I+100*G+10*H+T;
                  c1 = (mod(n,E)==I);
                  c2 = (mod(n,I)==G);
                  c3 = (mod(n,G)==H);
                  c4 = (mod(n,H)==0);
                  c5 = (mod(n,T)==0);
                  if (all([c1 c2 c3 c4 c5]))
                     n
                  end
               end
            end
         end
      end
   end

Output: 

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n = 87215
n = 94315

Janardhanan Sivaramakrishnan - 3 years, 5 months ago
Md Mehedi Hasan
Dec 11, 2017

E I G H T ‾ \overline{\color{#20A900}E\color{#3D99F6}I\color{#D61F06}G\color{#E81990}H\color{#BA33D6}T} represented by only two number. They are 87215 87215 and 94315 94315 which follow above condition.

So the sum is 87215 + 94315 = 181530 87215+94315=\boxed{181530}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

But how did u found it

Abhinav Shripad - 3 years, 5 months ago

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