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?

Let's call the number, N, and the digit unit, U.

So, N is formed with a string of 49 (say a's), ending in U. (0 =< a =<9).

(N - U)/10 = (string of 49 a's)

(N - U) = 10 *  (string of 49 a's)

49 a's sum to 49*a, so is congruent to a mod 49.

Taking mod 49 :

(N - U) (is congruent to)  a (mod 49)

Since N = 239 * K ==> N = (5*49 + 4)*K

4*K - U (is congruent to) a (mod 49)

U (is congruent to 4*K - a) (mod 49)

A congruence to 49, is also a congruence to 7. (49 = 7 x 7)

U (is congruent to 4*K - a) (mod 7)

So U =< 6.

Posted by pcbouhid on 2005-05-13 17:10:30