MHB What Are the Last Two Digits of the Sum of Factorials from 7! to 2016!?

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The discussion revolves around finding the last two digits of the sum of factorials from 7! to 2016!. Participants engage in solving the problem, with several members successfully providing correct solutions. Topsquark is highlighted for their solution, along with other contributors like kaliprasad and lfdahl. The thread emphasizes the importance of following guidelines for problem-solving submissions. Ultimately, the focus remains on the mathematical challenge presented by the factorial sum.
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Here is this week's POTW:

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What are the last two digits in $7! + 8! + 9! + ... + 2016 !$?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to the following members for their correct solution::)

1. topsquark
2. kaliprasad
3. lfdahl

Solution from topsquark:
Since 10!, ... , 2016! are all multiples of 100 all we need to worry about are 7!, 8!, and 9! for the last two digits. Since 7! + 8! + 9! = 7! (1 + 8 + 8*9) = 5040*81 we have [math]5040 \cdot 81 \equiv 40 \cdot 81 \equiv 3240 \equiv 40 \text{ mod(100)}[/math] the last two digits must be 40.

(Or you could just calculate 7! + 8! + 9! = 408240.)
 
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