The product of two consecutive positive integers is divisible by 2.

State True/False.

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True

False

answer is A.

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Detailed Solution

It is given two consecutive positive integers.
Any two consecutive positive integers can be represented as (n-1) and n or 2q+1 and 2q, where q is a whole number.
Let two consecutive positive integers be (n-1) and n.
Then, their product is

n (n − 1) = n 2 − n

……(1)
A positive integer can be represented in the form 2q+1.
Then equation (1) becomes,

n 2 − n = 2 q + 1 2 − 2 q + 1 n 2 − n = (4 q 2 + 4 q + 1 − 2 q − 1)

n 2 − n = 2 (2 q 2 + q) ......... (2)

Therefore (2) is a multiple of 2.
Equation (2) can be rewrite as,

n 2 − n = (2 q) 2 + (2 q) .......... (3)

Therefore (3) is a multiple of 2.
Since equation (2) and equation (3) are divisible by 2, their products will also be divisible by 2.
Therefore it is true that the product of two consecutive positive integers is divisible by 2.
Hence the correct option is 1.

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