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

 Oct 6, 2018
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\(\text{apparently }\sum \limits_{k=1}^n ~k! \cdot k = (n+1)! - 1 \\ 0!=1 \\ \text{so the expression shown }=(n+1)!-1+1 = (n+1)! \\ \text{so the largest prime }<(n+1) \text{ is what we're after}\\ \text{and this is }47.\)

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 Oct 6, 2018

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