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

  • MHB
  • Thread starter anemone
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    2016
In summary, the problem statement for POTW #243 is to find the last two digits of a series of factorials, given a specific pattern of numbers. Possible approaches to solving this problem include manually calculating the factorials and finding the pattern, or using a calculator or programming language. Finding the last two digits can provide valuable information about the pattern and behavior of the factorials, and can be used in combination with other techniques for solving more complex mathematical problems. Various tools and techniques, such as programming languages and mathematical tools, can be used to solve this problem. By solving this problem, one can improve their mathematical and problem-solving skills and gain a deeper understanding of factorial patterns and their real-world applications.
  • #1
anemone
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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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  • #2
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 \(\displaystyle 5040 \cdot 81 \equiv 40 \cdot 81 \equiv 3240 \equiv 40 \text{ mod(100)}\) the last two digits must be 40.

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

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

What is the problem statement for POTW #243?

The problem is to find the last two digits of a series of factorials, given a specific pattern of numbers.

What are some possible approaches to solving this problem?

One possible approach is to manually calculate the factorials and then find the pattern in the last two digits. Another approach is to use a calculator or programming language to automate the process.

What is the significance of finding the last two digits in a series of factorials?

The last two digits can provide valuable information about the overall pattern and behavior of the factorials. They can also be used in combination with other techniques to solve more complex mathematical problems.

Are there any specific tools or techniques that can be used to solve this problem?

There are various programming languages and mathematical tools that can be used to solve this problem, such as Python, Java, or even a simple spreadsheet program like Excel.

What can be learned from solving this problem?

Solving this problem can improve one's mathematical and problem-solving skills, as well as deepen understanding of factorial patterns and their applications in real-world situations.

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