Math Problem Statement

Prove the square of any odd integer n has the form 8m + 1 for some integer m.

Solution

The problem asks to prove that the square of any odd integer nn has the form 8m+18m + 1 for some integer mm. Let's go through the steps shown in the image and fill in the blanks as needed.

Step 1: Division Algorithm

For any integer nn, it can be expressed as: n=4q+rwith0r<4n = 4q + r \quad \text{with} \quad 0 \leq r < 4

Step 2: Possible Values of rr

  • (a) If r=0r = 0, then n=4qn = 4q.
  • (b) If r=1r = 1, then n=4q+1n = 4q + 1.
  • (c) If r=2r = 2, then n=4q+2n = 4q + 2.
  • (d) If r=3r = 3, then n=4q+3n = 4q + 3.

Step 3: Ignoring Even nn Cases

Since we are concerned with odd integers nn, we can ignore cases (a) and (c):

  • Case (a): n=4q=2(2q)n = 4q = 2(2q), which is even.
  • Case (c): n=4q+2=2(2q+1)n = 4q + 2 = 2(2q + 1), which is also even.

This means nn is even in both cases, so they don't apply to our situation.

Step 4: Consider Odd Cases

Now consider the cases where nn is odd, i.e., cases (b) and (d).

  • Case (b): n=4q+1n = 4q + 1 Then: n2=(4q+1)2=16q2+8q+1=8(2q2+q)+1n^2 = (4q + 1)^2 = 16q^2 + 8q + 1 = 8(2q^2 + q) + 1 Here, m=2q2+qm = 2q^2 + q, so n2=8m+1n^2 = 8m + 1.

  • Case (d): n=4q+3n = 4q + 3 Then: n2=(4q+3)2=16q2+24q+9=8(2q2+3q+1)+1n^2 = (4q + 3)^2 = 16q^2 + 24q + 9 = 8(2q^2 + 3q + 1) + 1 Here, m=2q2+3q+1m = 2q^2 + 3q + 1, so n2=8m+1n^2 = 8m + 1.

Conclusion

In both cases where nn is odd, n2n^2 has the form 8m+18m + 1. Thus, the claim is proved.


Would you like more details on any part of this explanation? Here are five related questions you could consider:

  1. How can we use this method to prove properties of other forms of integers?
  2. What happens if we consider different modulo conditions instead of n=4q+rn = 4q + r?
  3. Can you show how this proof method works for other cases, like n2=4m+1n^2 = 4m + 1 for even numbers?
  4. Why is the choice of rr limited to 0, 1, 2, and 3 in this context?
  5. 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