When you square an odd number is it odd?

Yes, when you square an odd number, the result is always an odd number. This is a consistent property in mathematics, holding true for all odd integers.

Understanding Odd Numbers

An odd number is any integer that cannot be exactly divided by 2. This means it will always leave a remainder of 1 when divided by 2. Examples include 1, 3, 5, 7, 9, and so on. Mathematically, an odd number can be represented in the form $2n + 1$, where $n$ is any integer.

The Rule: Squaring an Odd Number

As confirmed by Homework.Study.com, "The answer to your question is yes, an odd number squared will always return another odd number." This means that no matter which odd number you choose, multiplying it by itself will always yield an odd product.

Why This Works Mathematically

We can demonstrate this property using a simple algebraic proof:

1. Represent an Odd Number: Let any odd number be represented as $2n + 1$, where $n$ is an integer (e.g., if $n=1$, the number is $2(1)+1 = 3$; if $n=2$, the number is $2(2)+1 = 5$).
2. Square the Odd Number: Now, let's square this general form:
   $(2n + 1)^2$
3. Expand the Expression: Using the formula $(a+b)^2 = a^2 + 2ab + b^2$:
   $(2n + 1)^2 = (2n)^2 + 2(2n)(1) + (1)^2$
   $= 4n^2 + 4n + 1$
4. Factor Out a 2: We can factor out a 2 from the first two terms:
   $= 2(2n^2 + 2n) + 1$

Since $2n^2 + 2n$ is an integer, let's call it $k$. Then the expression becomes $2k + 1$. Any number that can be written in the form $2k + 1$ is, by definition, an odd number. Therefore, squaring an odd number always results in another odd number.

Practical Examples

Let's look at a few examples to illustrate this property:

As these examples show, the square of an odd number consistently yields an odd number.