prove that the product of two consecutive positive integers is divisible by 2
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let the two consecutive integers be x and x+1 so their product is x^2+x
if x is positive that 2k then,
(2k)^2+2k is divisible and
if x is negative that is 2k+1...
but substitution in x^2+2 we get that it is also divisible by 2
hence proved
if x is positive that 2k then,
(2k)^2+2k is divisible and
if x is negative that is 2k+1...
but substitution in x^2+2 we get that it is also divisible by 2
hence proved
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Let n and n + 1 are two consecutive positive integer
We know that n is of the form n = 2q and n + 1 = 2q + 1
n (n + 1) = 2q (2q + 1) = 2 (2q2 + q)
Which is divisible by 2
If n = 2q + 1, then
n (n + 1) = (2q + 1) (2q + 2)
= (2q + 1) x 2(q + 1)
= 2(2q + 1)(q + 1)
Which is also divisible by 2
Hence the product of two consecutive positive integers is divisible by 2
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