Question

-Prove that the product of three consecutive positive
intogers is divisible by 6.​

Answer 1

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Prove that the product of three consecutive positive integers is divisible by 6.

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Let us firstly find the factors of 6

6 = 2 × 3

➷➷it means the number which is divisible by 2 and 3 both will be divisible by 6.❖❖

NOW,

Let,

(x-1), (x), (x+1) are three consecutive positive integers then,

❖there are two case that x could be odd or even

so,

case (1)

when x is odd

then,

(x-1) will be even

and also

(x+1) will be even

it means their product will be divisible by 2.

and,

as we know that in any three consecutive positive integers one of them is divisible by 3

hence,

the product will also be divisible by 3

so, the product will be divisible by 2×3=6.

Case (2)

when x is even

then their product will be divisible by 2

(because one of the three consecutive positive integers is even)

and

one of the three will also be divisible by 3

hence,

the product of three consecutive positive integers will be divisible by 2×3=6.

❖Hence proved. ❖