- Home/
- SSC & Railways/
- Article
Prove that the Product of Three Consecutive Positive Integers is Divisible by 6
By BYJU'S Exam Prep
Updated on: September 25th, 2023
In the quest to prove the divisibility of the product of three consecutive positive integers by 6, we explore the fascinating pattern that emerges when we consider these numbers as n, n+1, and n+2. By examining the divisibility properties of consecutive integers, we can demonstrate that their product is divisible by both 2 and 3, two key factors in the divisibility rule for 6.
Table of content
Product of Three Consecutive Positive Integers is Divisible by 6
If three consecutive numbers n, n + 1, and n + 2 are used, then.
To show that their product is divisible by 6, we need to demonstrate that it is divisible by both 2 and 3.
When a number is divided by 3, the result is always one of the following: 0, 1, or 2.
Let n equal 3p, 3p + 1, or 3p + 2, with p being an 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.
Similar to this, whenever a number is divided by 2, the result is either 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.
What are Consecutive Integers?
Consecutive integers are a sequence of numbers that follow each other in order without any gaps. In this sequence, each number is exactly one unit higher than the previous number. For example, the consecutive integers starting from 1 would be 1, 2, 3, 4, 5, and so on. Similarly, the consecutive integers starting from -3 would be -3, -2, -1, 0, 1, 2, and so forth.
Summary:
Prove that the Product of Three Consecutive Positive Integers is Divisible by 6
The result of three successive positive numbers n (n + 1) (n + 2) is shown to be divisible by 6. Understanding the divisibility properties of consecutive integers allows us to make deductions and solve equations more efficiently.
Related Questions:
- Calculate the Molecular Mass of Co No32 6h2o
- What is the smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case
- Name the Animals Which Can Be Seen 1 on the Branches 2 on the Trunk 3 on the Leaves 4 on the Ground and 5 Around the Tree
- a2 + b2 is Equal to
- Explain How a Potential Barrier is Developed in a P N Junction Diode
- Find the Focal Length of a Lens of Power 2 0 D What Type of Lens is This
- Find the Missing Number 0 7 26 63