Question

The sum of two consecutive odd numbers is always divisible by 4. Verify this statement with the help of some examples.

Answer 1

Hint: We are asked to verify the given statement- ‘the sum of two consecutive odd numbers is always divisible by 4’. We start to solve the problem by finding out the divisibility rule for the number 4. Then, we verify if the sum of odd numbers is always divisible by 4.

Complete step-by-step solution:
We are asked to verify the given statement – ‘the sum of two consecutive odd numbers is always divisible by 4’. We will be solving the given question using the divisibility rule for the number 4.
A divisibility rule is an easy way to identify whether a given number is divisible by a fixed number without having to perform the long division.
The divisibility rule for the number 4 is given as follows,
The given number is divisible by 4 if the number formed by the last two digits of the given number is divisible by 4.
Example : 412
The last two digits of the number are 12. As the number 12 is divisible by 4, the number 412 is also divisible by 4.
The odd numbers end with digits 1,3,5,7,9. The numbers are not exactly divisible by 2 .
Example: 1, 3, 5, 7.
Now,
Let us verify the given statement using some examples.
We need to consider any two consecutive odd numbers. The consecutive odd numbers are given as follows,
$\Rightarrow \left( 1,3 \right),\left( 3,5 \right),\left( 5,7 \right),\left( 7,9 \right)$
The sum of the consecutive odd numbers 1 and 3 results in 4. It is given by
$\Rightarrow 1+3=4$
The number formed by the last two digits of the number 4 is 04. As the number 04 is divisible by 4, the number 4 is also divisible by 4.
 The sum of the consecutive odd numbers 3 and 5 results in 8. It is given by
$\Rightarrow 3+5=8$
The number formed by the last two digits of the number 8 is 08. As the number 08 is divisible by 4, the number 8 is also divisible by 4.
The sum of the consecutive odd numbers 5 and 7 results in 12. It is given by
$\Rightarrow 5+7=12$
The number formed by the last two digits of the number 12 is 12. As the number 12 is divisible by 4, the number 12 is also divisible by 4.
The sum of the consecutive odd numbers 7 and 9 results in 16. It is given by
$\Rightarrow 9+7=16$
The number formed by the last two digits of the number 16 is 16. As the number 16 is divisible by 4, the number 16 is also divisible by 4.
From the above,
$\therefore$ The sum of two consecutive odd numbers is always divisible by 4.
Hence, verified.

Note: The word ‘consecutive’ refers to the numbers that follow each other continuously from the smallest to the largest. In simple terms, it refers to successive terms. For example, 3, 4, 6, 7 are consecutive numbers. The consecutive odd numbers are $\left( 2n+1 \right),\left( 2n+3 \right),\left( 2n+5 \right),\left( 2n+7 \right)...$ where n = 0,1,2….