In 510909x21y1709440000 the given odd positions, starting counting at the left with 5 in position 1, add up to 11 and the given even positions add up to 41, for a total of 52. The difference between all the even digits and all the odd digits must be a multiple of 11 as 21! is a multiple of 11. Before x and y are filled in, the difference is 4111=30, so we need to increase the difference by 3 mod 11. We can't increase the difference by 14 or more, so the difference yx must be 3.
21! is also divisible by 9, so the sum of the digits must be congruent to 0 mod 9. Without the x and y the total is 52, which is congruent to 7, so we need to add 2 to the sum mod 9.
The only pair of digits that satisfy both of these is (x,y) = (4,7).

Posted by Charlie
on 20130701 14:23:57 