Find the largest prime number that divides the quantity \(0! + (1!) \times 1 + (2!) \times 2 + (3!) \times 3 + \cdots + (50!) \times 50\) .
+129973 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
