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Apr 27, 2015

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

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

Raghunath Reddy

Member since Apr 11, 2014

 Sol; Let us three consecutive  integers be, n, n + 1 and n + 2. Whenever a number is divided by 3 the remainder obtained is either 0 or 1 or 2. let 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, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3. If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3. So that n, n + 1 and n + 2 is always divisible by 3. â‡’ n (n + 1) (n + 2) is divisible by 3.   Similarly, whenever a number is divided 2 we will get the remainder is 0 or 1. âˆ´ n = 2q or 2q + 1, where q is some integer. If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2. If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2. So that n, n + 1 and n + 2 is always divisible by 2. â‡’ n (n + 1) (n + 2) is divisible by 2.   But n (n + 1) (n + 2) is divisible by 2 and 3.   âˆ´ n (n + 1) (n + 2) is divisible by 6.

Syeda

Member since Jan 25, 2017

Answer. Let us three consecutive  integers be, n, n + 1 and n + 2. Whenever a number is divided by 3 the remainder obtained is either 0 or 1 or 2. let 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, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3. If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3. So that n, n + 1 and n + 2 is always divisible by 3. â‡’ n (n + 1) (n + 2) is divisible by 3.   Similarly, whenever a number is divided 2 we will get the remainder is 0 or 1. âˆ´ n = 2q or 2q + 1, where q is some integer. If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2. If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2. So that n, n + 1 and n + 2 is always divisible by 2. â‡’ n (n + 1) (n + 2) is divisible by 2.   But n (n + 1) (n + 2) is divisible by 2 and 3.   âˆ´ n (n + 1) (n + 2) is divisible by 6
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