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Find the largest prime number that divides the quantity \(0! + (1!) \times 1 + (2!) \times 2 + (3!) \times 3 + \cdots + (50!) \times 50\) .

 Nov 23, 2018
 #1
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0!   + 1*1! + 2*2!   + 3*3!   +  4*4!  + 5*5!  .......+ 50*50!

 

One of our members, Heureka, can probably prove this in a more elegant manner....but....here's my best attempt

 

Note the first few terms

 

1  + 1^2 *0! + 2^2 * 1!   =   6     = 3!     

 

1 + 1^2*0! + 2^2 * 1!  + 3^2*2!   =  24   =  4!

 

1 + 1^2*0! + 2^2 * 1!  + 3^2*2!  + 4^2* 3!     = 120    =  5!       and

 

1 + 1^2 *0! + 2^2 * 1!  + 3^2^2!  + 4^2* 3! + 5^2*4! = 6!

 

So....it would appear that we can write the sum as  

 

1   + 1*1! + 2*2!   + 3*3!   +  4*4!  + 5*5!  .......+ (n)^2 * (n - 1)!!   = (n + 1)!

 

So

 

0!   + 1*1! + 2*2!   + 3*3!   +  4*4!  + 5*5!  + .......+ 50*50!    =

 

1 + 1^2 *0! + 2^2 * 1!   + 3^2^2!  + 4^2* 3! + 5^2*4 +  ....+ 50^2 * 49! = ( 51) ! 

 

And largest prime that would divide this   is    47

 

 

cool cool cool

 Nov 23, 2018
edited by CPhill  Nov 23, 2018
edited by CPhill  Nov 23, 2018
edited by CPhill  Nov 23, 2018
 #2
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deleted.

 Nov 23, 2018
edited by Guest  Nov 23, 2018
edited by Guest  Nov 23, 2018
edited by Guest  Nov 23, 2018
edited by Guest  Nov 23, 2018
edited by Guest  Nov 23, 2018

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