Math Problem Statement
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Solution
To solve this problem, let’s analyze the mathematical properties of the sum of squares of two consecutive even numbers.
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Define consecutive even numbers:
Let the first even number be , where is even. The next consecutive even number is . -
Sum of squares:
The sum of the squares of these numbers is:
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Expand the expression:
Expanding :
Combine like terms:
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Factorize:
Factor out :
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Divisibility analysis:
- The expression is clearly divisible by 2.
- Test for higher divisibility by substituting values for :
- If : .
- If : .
- Clearly, the sum of squares is always a multiple of 2, 4, and other factors.
Correct Answers:
- The sum of squares is always divisible by 2, 4, and 8.
Let me know if you'd like me to go into more detail or clarify further!
Questions for exploration:
- Can you prove the result for other pairs of even numbers?
- What happens if the numbers are consecutive odd numbers instead?
- Can you derive similar results for three consecutive numbers?
- Is there any case where the sum of squares is divisible by 16? Why or why not?
- How does this extend to numbers of different modular forms (e.g., )?
Tip: Always simplify expressions and factorize them to reveal divisibility properties!
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Math Problem Analysis
Mathematical Concepts
Algebra
Number Theory
Consecutive Even Numbers
Divisibility
Formulas
Sum of squares formula: n^2 + (n + 2)^2 = 2n^2 + 4n + 4
Factored form: 2(n^2 + 2n + 2)
Theorems
Divisibility by 2 and 4
Properties of even numbers
Suitable Grade Level
Grades 9-12
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