The square of every odd integer is of the form

Question 4Answer

a.

8k+1,k∈Z8k+1,k∈Z

b.

4k+1,k∈Z4k+1,k∈Z


c.

3k+1,k∈Z3k+1,k∈Z


d.

4k+3,k∈Z


To solve the problem, let's analyze the square of an odd integer and see which of the given forms it matches.

### Step 1: Express an Odd Integer
An odd integer can be expressed as:
\[ n = 2m + 1 \]
where \( m \) is an integer.

### Step 2: Square the Odd Integer
Let's square \( n \):
\[
n^2 = (2m + 1)^2 = 4m^2 + 4m + 1 = 4(m^2 + m) + 1
\]
Notice that \( m^2 + m \) is an integer, so let's set \( k = m^2 + m \). Therefore, we can write:
\[
n^2 = 4k + 1
\]
where \( k \) is an integer.

### Conclusion:
The square of every odd integer is of the form \( 4k + 1 \), where \( k \) is an integer.

Thus, the correct answer is:

**b. \( 4k + 1, \ k \in \mathbb{Z} \)**

Would you like any further explanation on this topic or have any questions?

Here are five related questions for further practice:
1. What is the general form for the square of an even integer?
2. Prove that the square of an odd integer is always congruent to 1 modulo 4.
3. If \( n = 5 \), what is the value of \( n^2 \) and how does it fit into the form \( 4k + 1 \)?
4. Is the statement "The square of any integer is of the form \( 3k \), \( k \in \mathbb{Z} \)" true or false?
5. What is the square of the odd integer \( 7 \), and into what form does it fit from the options provided?

**Tip:** Whenever you're given a question involving odd or even integers, remember to express them as \( 2m + 1 \) or \( 2m \) to simplify the problem.

Explore the mathematical form of squares of odd integers. Learn how every square of an odd integer fits the form 4k + 1, where k is an integer. This problem involves concepts from number theory and modular arithmetic.

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