21!=510909x21y1709440000

Without calculating 21!, what are the digits marked x and y?

Since we know 9 is a factor we can make the digit sum a multiple of 9: 52+x+y
The only attainable sums are 54 and 63
So x+y=2 or x+y=11

Since we know 11 is a factor we can make the differences of the even and odd numbered digits a multiple of 11:
Odds 41+y
Evens 11+x
Difference 30+y-x
The only attainable differences are 22 and 33
So x-y=8 or x-y=-3

There are four combinations to check.
Two won't give integer solutions.

x+y=2 and x-y=8 has a negative solution.

x+y=11 and x-y=-3 has the only viable solution x=4, y=7

A quick web search confirms
21! = 51090942171709440000

Posted by Jer on 2013-07-01 11:20:38