##### Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back

clear

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. ⇒ 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. ∴ 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. ⇒ n (n + 1) (n + 2) is divisible by 2.   But n (n + 1) (n + 2) is divisible by 2 and 3.   ∴ 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. ⇒ 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. ∴ 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. ⇒ n (n + 1) (n + 2) is divisible by 2.   But n (n + 1) (n + 2) is divisible by 2 and 3.   ∴ n (n + 1) (n + 2) is divisible by 6
###### Like NextGurukul? Also explore our advanced self-learning solution LearnNext
Offered for classes 6-12, LearnNext is a popular self-learning solution for students who strive for excellence
Explore
Animated Video
lessons
All India
Test Series
Interactive Video
Experiments
Best-in class
books