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“The product of three consecutive positive integers is divisible by 6”. Is this statement true or false? Justify your answer

Solution:

True,

n, (n + 1) and (n + 2) are the three consecutive integers

Here one integer of these three should be divisible by 2 and the other one should be divisible by 3

Therefore, the product of three consecutive integers is divisible by 6

✦ Try This: Prove that the product of 3 consecutive positive integers is divisible by 6

Consider n, n + 1 and n + 2 as the three consecutive positive integers.

When a number is divided by 3, the remainder obtained is either 0 or 1 or 2.

n = 3p or 3p + 1 or 3p + 2, where p is some integer.

If n = 3p, then n is divisible by 3

If n = 3p + 1, ⇒ n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.

If n = 3p + 2, ⇒ n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.

We can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 3.

n (n + 1) (n + 2) is divisible by 3.

When a number is divided 2, the remainder obtained is 0 or 1

n = 2q or 2q + 1, where q is some integer.

If n = 2q ⇒ n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2.

If n = 2q + 1 ⇒ n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.

We can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 2.

n (n + 1) (n + 2) is divisible by 2

Therefore, n (n + 1) (n + 2) is divisible by 2 and 3.

n (n + 1) (n + 2) is divisible by 6

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1

NCERT Exemplar Class 10 Maths Exercise 1.2 Problem 3

Summary:

The product of three consecutive integers is divisible by 6. Hence the given statement is true.

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