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?
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.
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Posted by Charlie
on 2005-05-13 18:00:09 |
A systematic approach
