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50 - Digit Number II (Posted on 2005-05-13) Difficulty: 2 of 5
I am thinking of a fifty-digit number divisible by 239, of which, each digit is the same, except the ones digit. What is the ones digit?

See The Solution Submitted by Dustin    
Rating: 2.8000 (5 votes)

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Solution A systematic approach | Comment 4 of 11 |

The following table shows the mod 239 values of rep-unit numbers up to 20 repetitions of the digit 1:

1       1
11      11
111     111
155     1111
117     11111
215     111111
0       1111111
1       11111111
11      111111111
111     1111111111
155     11111111111
117     111111111111
215     1111111111111
0       11111111111111
1       111111111111111
11      1111111111111111
111     11111111111111111
155     111111111111111111
117     1111111111111111111
215     11111111111111111111
0       111111111111111111111

Every multiple of 7 repeated 1's has value 0 mod 239.  Ten times zero is zero, so multiplying a 49-digits of all 1's by 10 results in zero mod 239.

If 1's didn't work we'd have to try 2's, 3's, etc.

An alternative method:

Alternatively, with a large precision computation, a 50-digit rep-unit number is shown to be congruent to 1 mod 239.  So changing the repunits to repetitions of any other digit (or just changing the last digit higher) would result in a modular value of 2 through 9, so the only choice is to subtract 1, giving the same answer as described before: 49 1's and a zero.


  Posted by Charlie on 2005-05-13 18:00:09
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