Prove that the product of two …
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Sia ? 6 years, 4 months ago
Let n-1 and n be consecutive positive integers,Let P be their product
Then {tex}\style{font-family:Arial}{\style{font-size:12px}{\mathrm n(\mathrm n-1)=\mathrm n^2-1\;.................(1)}}{/tex}
We know that any positive integers is of the form 2q or 2q + 1, where q is a positive integer
Case I: When n = 2q, then
P=n2 - n = (2q)2 - 2q = 4q2 - 2q = 2q(2q - 1)-----(2)
Case II: When n = 2q + 1, then
P=n2 - n = (2q + 1)2 - (2q + 1)
= 4q2 + 4q + 1 - 2q - 1
= 4q2 + 2q
P = 2q(2q + 1)..........(3)
From (2) and (3) we conclude that the product of n-1 and n is divisible by 2
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