find the remainder

Number Theory Level 2

The remainder when 1 ! + 2 ! + 3 ! + 4 ! . . . . . . . . . . + 2013 ! 1!+2!+3!+4!..........+2013! divided by 24 24 is


The answer is 9.

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Govindarajan Prakash by govindarajan prakash , 20, India

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2 solutions

1!=1(mod24) 2!=2(mod 24) 3!=6(mod 24) 4!=0(mod24) ''''''''''''''''''''''''' ''''''''''''''''''''''''' 2013!=0(mod 24) hence1!+2!+3!+4!..........+2013!= 1+2+6+0=9

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Vibha Sharma
Nov 1, 2015

24 = 1 * 2 * 3 * 4 = 4!

(1 ! + 2! + 3! +4! +5! + ........2013 !) /4!

=(1! + 2! + 3! ) /4! + (4! + 5!........2013!) /4!

=(1 + 2 + 6) / 4! + 0

= 9/ 4!

= 9 bcz 9 < 4!

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