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 41-11=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 y-x 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 2013-07-01 14:23:57