Math Problem Statement
Solution
The problem asks to prove that the square of any odd integer has the form for some integer . Let's go through the steps shown in the image and fill in the blanks as needed.
Step 1: Division Algorithm
For any integer , it can be expressed as:
Step 2: Possible Values of
- (a) If , then .
- (b) If , then .
- (c) If , then .
- (d) If , then .
Step 3: Ignoring Even Cases
Since we are concerned with odd integers , we can ignore cases (a) and (c):
- Case (a): , which is even.
- Case (c): , which is also even.
This means is even in both cases, so they don't apply to our situation.
Step 4: Consider Odd Cases
Now consider the cases where is odd, i.e., cases (b) and (d).
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Case (b): Then: Here, , so .
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Case (d): Then: Here, , so .
Conclusion
In both cases where is odd, has the form . Thus, the claim is proved.
Would you like more details on any part of this explanation? Here are five related questions you could consider:
- How can we use this method to prove properties of other forms of integers?
- What happens if we consider different modulo conditions instead of ?
- Can you show how this proof method works for other cases, like for even numbers?
- Why is the choice of limited to 0, 1, 2, and 3 in this context?
- How does this proof generalize to different mathematical structures, like polynomials?
Tip: When proving properties about squares of integers, consider the division algorithm and modular arithmetic for simplification.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Division Algorithm
Modular Arithmetic
Formulas
n = 4q + r, 0 <= r < 4
n^2 = 8m + 1
Theorems
Division Algorithm
Suitable Grade Level
Undergraduate Mathematics
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