State true or false:
The product of three consecutive positive integers is divisible by 6.

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True

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answer is A.

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Detailed Solution

We need to find whether the product of three consecutive positive integers is divisible by 6.
So, assign the two consecutive positive integers as

n, (n + 1) and (n + 2)

.
Hence, their product is

n (n + 1) (n + 2)

……. (1)
Use the property that a positive integer can be represented in the form

6 q + 1

, and substitute it in equation (1). Hence,

n (n + 1) (n + 2) = (6 q + 1) (6 q + 2) (6 q + 3)

\Rightarrow n (n + 1) (n + 2) = 6 [(6 q + 1) (3 q + 1) (2 q + 1)]

……. (2)
Hence, equation (2) is divisible by 6.
Use the property that a positive integer can be represented in the form 6q, and substitute it in equation (1). Hence,

n (n + 1) (n + 2) = 6 q (6 q + 1) (6 q + 2)

\Rightarrow n (n + 1) (n + 2) = 6 [(6 q + 1) (6 q + 1)]

……... (3)
Hence, equation (3) is divisible by 6.
Use the property that a positive integer can be represented in the form

6 q + 2

, and substitute it in equation (1).

n (n + 1) (n + 2) = (6 q + 2) (6 q + 3) (6 q + 4)

\Rightarrow n (n + 1) (n + 2) = 6 [(3 q + 1) (2 q + 1) (6 q + 4)]

…….. (4)
Hence, equation (4) is divisible by 6.
As equation (2), (3) and (4) are divisible by 6, their product will be divisible by 6 as well.
Therefore, it is true that the product of three consecutive positive integers is divisible by 6.
Hence, option 1 is correct.

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