Logical Nightmare

Number Theory Level 2

Find the last digit of ( 1 ! + 2 ! + . . . . . . . . 2009 ! ) 500 { (1!+2!+........2009!) }^{ 500 }


The answer is 1.

Vighnesh Raut by Vighnesh Raut , 23, India

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3 solutions

Astro Enthusiast
Aug 2, 2014

From 5! onwards, all numbers end with zeroes. So compute for 1 ! + 2 ! + 3 ! + 4 ! 1!+2!+3!+4! which results to 33. Now 3 3 500 33^{500} mod10 is 1. :D

3 Helpful 0 Interesting 0 Brilliant 0 Confused

@Precious Prestosa -Are you sure that the answer to your new problem "the last 3!" is correct?

Jayakumar Krishnan - 6 years, 10 months ago

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Yes :) Why? :o

Astro Enthusiast - 6 years, 10 months ago

Oh no! I'm sorry, I already messaged brilliant to change it. Thank you for pointing it out :)

Astro Enthusiast - 6 years, 10 months ago
William Isoroku
Aug 2, 2014

Any number above 4! Contains a 0 in its units digit. So add the first four factorials and the units digit is 3. Now 3 raised to any positive integral multiples of 4 has 1 in its units digit. Luckily 500 is divisible by 4.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Nitish Deshpande
Jul 16, 2014

add units place till 4! because from 5! and onwards all terms end in 0 so upto 4! units place is3 now 3 raised to 500 units place is 1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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