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Answer this ,
"The product of three consecutive positive integers is divisible by 6 ". Is this statement true or false ? Justify your answer.
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Answers
Explanation:
Let the Product of three consecutive integers be k = n(n + 1)(n + 2).
(i) Divisible by 3:
Any positive integer is in the form of 3q, 3q + 1 and 3q + 2.
(a) When n = 3q, then k = 3q(3q + 1)(3q + 2) is divisible by 3.
(b) When n = 3q + 1, then k = (3q + 1)(3q + 2)(3q + 3)
= 3(3q + 1)(3q + 2)(q + 1) is divisible by 3.
(c) When n = 3q + 2, then k = (3q + 2)(3q + 3)(3q + 4)
= 3(3q + 2)(q + 1)(3q + 4) is divisible by 3.
Thus, product of 3 consecutive integers is divisible by 3.
(ii) Divisible by 2:
Any positive integer is in the form of 2m and 2m + 1.
(a) When n = 2m, then k = 2m(2m + 1)(2m + 2) is divisible by 2
(b) When n = 2m + 1, then k = (2m + 1)(2m + 2)(2m + 3)
= 2(2m + 1)(m + 1)(2m + 3) is divisible by 2.
Thus, Product of 3 consecutive integers is divisible by 2.
Therefore, Product of three consecutive integers is divisible by 6.
Hope it helps!
Answer:
It is the correct answer.
Step-by-step explanation:
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