A number of 50 digits has all its digits equal to 1 except the 26th digit. If the number is divisible by 13, then find the digit in the 26th place.
(In reply to
a little more straightforward by Eric)
That 111111 is divisible by 13 is an excellent observation. So the 50digit number is the sum of multiples of 111111, i.e. multiples of 13, plus (10*x+1)*10^24 where x is the unknown digit. If 10*x+1 is a multiple of 13, we have a solution. Now 91=13*7. so we can use x=9. Also 10^24 is congruent to 1 mod 13 (10^4 is congruent to 3), so any multiplier of 10^24 must be congruent to 0 mod 13, so I think that 9 has to be the only possible value for x.

Posted by Richard
on 20031119 00:10:42 