# Consecutive Numbers

## What are Consecutive numbers?

They are a series of numbers in which each number is greater than the preceding number by 1.

E.g. 8, 9, 10, 11, …

We can also say that, they are basically an arithmetic progression (A.P.), with a common difference of 1.

## Properties of Consecutive Numbers

### Property 1: Number of Consecutive numbers

If {a, a + 1, a + 2, … , b} is a set of consecutive integers, then
Number of elements in the set = b – a + 1

##### Q. Find the number of elements in the set of consecutive natural numbers 12 to 96.

Explanation:

Number of elements = b – a + 1 = 96 – 12 + 1 = 85

### Property 2

If the number of items in a set of consecutive integers is odd, then:
the sum of all integers is always divisible by the total items present in the set

Example: in a set of three consecutive numbers: 7, 8, 9.
Sum = 7 + 8 + 9 = 24 (it is divisible by 3)

Example: in a set of five consecutive numbers: 9, 10, 11, 12, 13.
Sum = 9 + 10 + 11 + 12 + 13 = 55 (it is divisible by 5)

### Property 3

If the number of items in a set of consecutive integers is even, then:
the sum of all integers is never divisible by the total items present in the set

Example: in a set of four consecutive numbers: 7, 8, 9, 10.
Sum = 7 + 8 + 9 + 10 = 34 (it is not divisible by 4)

Example: in a set of six consecutive numbers: 9, 10, 11, 12, 13, 14.
Sum = 9 + 10 + 11 + 12 + 13 + 14 = 69 (it is not divisible by 6)

### Property 4

Product of any two consecutive integers is always even (i.e. divisible by 2).

This is because one of them must be even (and the other odd), e.g. (3, 4), (12, 13)

Two consecutive integers can be written as:
n and n - 1 or
n and n + 1

Hence, any number of the form n(n - 1) or n(n + 1) will always be even.

### Property 5

Product of any three consecutive integers is always divisible by 6.

This is because one of them must be even, and one of them must be divisible by 3, e.g. (3, 4, 5), (12, 13, 14)

Three consecutive integers can be written as: n – 1, n and n + 1

Hence, any number of the form (n - 1)n(n + 1) or n($n^2$ - 1) or ($n^3$ - n) will always be divisible by 6.

### Property 6

Out of any n consecutive integers, exactly one number will be divided by n.

Example:
Four consecutive integers: 14, 15, 16, 17
Only 16 is divisible by 4.

### Property 7

The product of n consecutive integers will be divisible by n!

Example:
Four consecutive integers: 14, 15, 16, 17
(14 × 15 × 16 × 17) is divisible by 4!

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