Determine the last two non-zero digits of 36! - 24!

Extra Challenge: Solving this puzzle without a computer program.

To get the factorial of 24 using only a calculator, you can multiply these numbers and restrict the result to mod 1000000:

3,4,6,7,8,9,
11,6,13,14,3,16,17,18,19,2,
21,22,23,24

Note that both 2 and 5 have been left out as together they contribute only a trailing zero. 10 has been left out as well for the same reason, and 12 has been replaced by 6, which together with 15 being replaced by 3, removes another trailing zero, as does the change from 20 to 2. This accounts for all the trailing zeros of 24!.

 1 * 3 = 3
   mod 1000000 = 3
* 4 = 12
   mod 1000000 = 12
* 6 = 72
   mod 1000000 = 72
* 7 = 504
   mod 1000000 = 504
* 8 = 4032
   mod 1000000 = 4032
* 9 = 36288
   mod 1000000 = 36288
* 11 = 399168
   mod 1000000 = 399168
* 6 = 2395008
   mod 1000000 = 395008
* 13 = 5135104
   mod 1000000 = 135104
* 14 = 1891456
   mod 1000000 = 891456
* 3 = 2674368
   mod 1000000 = 674368
* 16 = 10789888
   mod 1000000 = 789888
* 17 = 13428096
   mod 1000000 = 428096
* 18 = 7705728
   mod 1000000 = 705728
* 19 = 13408832
   mod 1000000 = 408832
* 2 = 817664
   mod 1000000 = 817664
* 21 = 17170944
   mod 1000000 = 170944
* 22 = 3760768
   mod 1000000 = 760768
* 23 = 17497664
   mod 1000000 = 497664
* 24 = 11943936
   mod 1000000 = 943936

The last six non-zero digits of 24! are 943936, and since 36! has four more trailing zeros than 24!, you can subtract 36 from 100 to get 64 as the last two non-zero digits of 36! - 24!.

A UBASIC calculation confirms 24! =  620448401733239439360000 and 36!-24! = 371993326789901216847551046417595760640000.

Posted by Charlie on 2014-01-12 17:23:08