Question

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

Answer 1

Answer:

product of three consecutive positive integer is divisible by 6

Explanation:

first take any three consecutive positive integers

such as 2,3,4.

then multiply three numbers

2*3*4=24

24 the is divisible by 2 and 3 then it is divisible by 6

therefore,24 in is divisible 6

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Answer 2

To prove:-

• The product of three consecutive positive integers is divisible by 6

Proof:-

• Let n be any positive integer.

Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5.

If n = 6q

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

• which is divisible by 6.

If n = 6q + 1

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

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

• Which is divisible by 6.

If n = 6q + 2

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

=> n(n + 1)(n + 2) = 12(3q + 1)(2q + 1)(2q + 3)

• Which is divisible by 6.

Similarly we can prove others.

Hence, it is proved that the product of three consecutive positive integers is divisible by 6.