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2006 AMC 10B Problem 11

Problem 11 of 25IntermediateNumber TheoryCounting & Probability

What is the tens digit in the sum 7!+8!+9!++2006!?7! + 8! + 9! + \cdots + 2006!\,?

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Solution

For n10,n\ge 10, n!n! is divisible by 100,100, so it does not affect the last two digits. The tens digit comes from 7!+8!+9!7!+8!+9! =5040+40320+362880=5040+40320+362880 =408240,=408240, whose tens digit is 4.4. Thus, the correct answer is C.

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Concepts: factorial · modular arithmetic · digits

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.