Solving A Factorial Equation

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[SG]

Here is my way, how I continued with k+20=k! If you subtract k from both sides you get 20=k!-k. ==> 20=k((k-1)!-1). That means since k is an integer, 20 must be divisible by k. That means k can only be 1;2;4;5;10 or 20. Testing these 6 numbers tells us, that k=4 is the only solution, means n=5.

[QSBraWQ]

Factorials have a cool sign. Thats the reason I watched this video

[echandler]

k+20 = k!
k divides both sides thus k must also divide 20.
1 ≤ k ≤ 20. 21 ≤ LHS ≤ 40 thus k = 4.

[toveirenestrand]

n^2 + 19n - n! = 0 = n[n + 19 - (n - 1)!], which means that n + 19 - (n - 1)! = 0. n = 4 does not fulfill the equation, but n = 5 does. So the answer is n = 5.