Question

The sum of 2 consecutive odd numbers is divisible by 4. Prove this statement by some examples.

Answer 1

Hint: Consider examples of 2 consecutive numbers which are odd. Consecutive numbers have a difference of 2 between them. Add the 2 consecutive odd numbers and prove that they are divisible by 4.

Complete Step-by-Step solution:
If x is any odd number, then x and x + 2 are consecutive odd numbers. Odd consecutive numbers are odd numbers that follow each other. They have a difference of 2 between every two numbers. If n is an odd number, then n, n + 2, n + 4 and n + 6 are odd consecutive numbers. Now let us use some examples and prove that the number we obtain is divisible by 4.
Let us consider some examples,
\[11+13=24\to \] 11 and 13 are consecutive odd numbers.
 \[9+11=20\to \] 9 and 11 are consecutive odd numbers.
\[25+27=52\to \] 25 and 27 are consecutive odd numbers.
\[103+105=208\to \] 103 and 105 are consecutive odd numbers.
Thus the sum we got 24, 20, 52 and 208 are divisible by 4.
Thus the sum of 2 consecutive odd numbers is divisible by 4.
An even number is divisible by 2, so it can be represented by 2n, where n is an integer. If we add 1 to an even number, then it will be added. Therefore, an odd number can be represented as 2n + 1.
If (2n + 1) is an odd number, then the next odd number will be (2n + 3). Therefore, the sum of two consecutive odd numbers can be represented as,
\[\left( 2n+1 \right)+\left( 2n+3 \right)=2n+2n+1+3\]
                                   \[=4n+4=4\left( n+1 \right)\]
Thus we have 4 (n + 1), where 4 is a factor of 4n +4. This means that 4n + 4 is divisible by 4.
Thus the sum of two consecutive odd numbers is divisible by 4.

Note: You can prove it by considering a few examples alone. Any consecutive odd numbers you take will yield an even number divisible by 4. Remember what a consecutive number. If we are adding 3 and 7, they both are odd but not consecutive. Their sum is 10, which is even but not divisible by 4.