Fear Factorial !

Algebra Level 2

Let p = 1 × 1 ! + 2 × 2 ! + 3 × 3 ! + 4 × 4 ! + … + 999 × 999 ! p= 1\times1! + 2\times2! + 3\times3! + 4\times4! + \ldots + 999\times 999!

Find the remainder when p + 1 p+1 is divided by 1000 ! 1000! .


The answer is 0.

Saurav Pal by Saurav Pal , 26, India

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3 solutions

Saurav Pal
Mar 26, 2014

Let's do by Hit and Trial Method.
Let p= 1.1! + 2.2! + 3.3! = 23
Then, x= 24
4! = 24
x/4! = 1
Remainder = 0.


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0

Saurav Sharma - 7 years, 2 months ago

p+1= 1+1.1!+2.2!+3.3!....999.999!
=1!(1+1) +2.2!+3.3!......999.999! =1!.2 +2.2!+3.3!....999.999! =2!+2.2!+3.3!......999.999! =2!(2+1) +3.3!...999.999! =2!.3 +3.3!...999.999! . . . =999!(999+1) =1000! p+1/1000! remainder 0

Raunak Tibrewala - 7 years, 2 months ago
Debarpan Adhikari
Mar 28, 2014

Simplifying the equation, we get the sum as p= (n+1)!-1. Hence, p= (999+1)!-1= 1000!-1. p+1= (1000!-1)+1= 1000!-1+1= 1000! 1000! factorial is obviously divisible by 1000! and hence leaving the remainder as 0.

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Sunil Bisoyi
Mar 28, 2014

Hint: 1+ 1!+ 2!+2!+ 3!+3!+3!+ ....

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