Let n = 2019, the least positive integer k

Q&ACategory: Exam HelpLet n = 2019, the least positive integer k
AY Sir Staff asked 12 months ago

Let n = 2019. The least positive integer k for which k(n²) (n² – 1²) (n² – 2²) (n² – 3²) … (n² – (n – 1)²) = r! for some positive integer ‘r’ is …

1 Answers
AY Sir Staff answered 12 months ago

Solution: We can rewrite the given expression as
k(n²) (n – 1) (n + 1) (n – 2) (n + 2) (n – 3) (n + 3)…(n + n – 1) (n – n + 1) = r!
=> (k) (n) (1) (2)… (n – 1) n (n + 1) (n + 2)… (2n – 1) = r!
=> k n (2n – 1)! = r!
In order to convert L.H.S. to a factorial, we require, k = 2
which will convert it into (2n)!
So the required value of k is 2.