Maths
Theorem: The product of any n consecutive positive integers is divisible by n! for n ≥ 1.
Proof: Let p be such a product, represented as
(1) p = k(k + 1)…(k + n - 1) for some k ε Z+
Then p can also be represented using factorials:
(2) p = (k + n - 1)!/(k - 1)!
Now let m = k + n - 1, and (2) can be rewritten as
(3) p = m!/(m - n)!
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Proof: Let p be such a product, represented as
(1) p = k(k + 1)…(k + n - 1) for some k ε Z+
Then p can also be represented using factorials:
(2) p = (k + n - 1)!/(k - 1)!
Now let m = k + n - 1, and (2) can be rewritten as
(3) p = m!/(m - n)!
To ( Read more... )
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