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12 digit number (Posted on 2007-01-22) Difficulty: 3 of 5
A 12 digit number is formed using these rules:

* You can only use digits 1,5 and 9 (base 10)
* The number is divisible by 37

Prove that the sum of the digits is not 76

No Solution Yet Submitted by atheron    
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Solution Analytic solution Comment 3 of 3 |
In a 12 digit number, each digit contributes its value to the sod;
and also it contributes to the mod 37 value of the whole number which depends only on its value and its position in the number; what power of 10 it is multiplied by.
If you consider the contribution to the mod 37 value of a 12 digit repunit, 111111111111, the contribution to mod 37 is:
    1,10,26,1,10,26,1,10,26,1,10,26
So for the 12 digit number to be 0 mod 37, the number of occurrences of each of 1, 5, and 9 would have to be a multiple of 3 in positions contributing 1, 10, and 26 to the mod value.
For example 111555999111 should be divisible by 37 (and it is).  Another example would be 159591915999.

Next:  count how many ways you can have 12 integers sum to 76 where each is either 1, 5, or 9.

1s 5s 9s
0 8 4
1 6 5
2 4 6
3 2 7
4 0 8

In none of these cases is each divisible by 3.
Therefore there are no such 12 digit numbers.


--------------------------------
for n1 in range(13):
    for n5 in range(13):
        for n9 in range(13):
            if n1 + 5*n5 + 9*n9 == 76  and  n1+n5+n9 == 12:
                print(n1,n5,n9)

for expo in range(12):
    print((10**expo)%37)
    

  Posted by Larry on 2023-11-16 10:47:46
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