To understand whether the sum of two consecutive odd numbers is always divisible by 4, let’s first define what we mean by consecutive odd numbers. Consecutive odd numbers can be expressed as:
Let the first odd number be n, then the next consecutive odd number will be n + 2.
Now, if we calculate the sum of these two consecutive odd numbers:
Sum = n + (n + 2) = 2n + 2
This simplifies to:
Sum = 2(n + 1)
This expression shows that the sum of two consecutive odd numbers is always an even number since it is 2 times something. Now, to determine whether this sum is divisible by 4, we need a couple of examples:
Example 1: Let’s take the first pair of consecutive odd numbers: 1 and 3.
Sum = 1 + 3 = 4. Since 4 is divisible by 4, this example holds true.
Example 2: Now, let’s take another pair: 5 and 7.
Sum = 5 + 7 = 12. The number 12 is not divisible by 4, which makes us rethink.
Example 3: Let’s look at the numbers 9 and 11.
Sum = 9 + 11 = 20, which is indeed divisible by 4.
Through these examples, we can start to notice a pattern. It turns out that while the sum of two consecutive odd numbers is always even, it is not always divisible by 4.
In conclusion, the statement that the sum of two consecutive odd numbers is always divisible by 4 is incorrect. It can be divisible by 4 in some cases, but there are also clear exceptions, as seen in the second example of adding 5 and 7.