Math Problem Statement

do

The sum of the squares of any two consecutive even numbers is always a multiple of which of the following? Select all of the correct answers: 2, 4, 6, 8, 10, 16.

Solution

To solve this problem, let’s analyze the mathematical properties of the sum of squares of two consecutive even numbers.

  1. Define consecutive even numbers:
    Let the first even number be nn, where nn is even. The next consecutive even number is n+2n+2.

  2. Sum of squares:
    The sum of the squares of these numbers is:
    n2+(n+2)2n^2 + (n+2)^2

  3. Expand the expression:
    Expanding (n+2)2(n+2)^2:
    n2+(n+2)2=n2+(n2+4n+4)n^2 + (n+2)^2 = n^2 + (n^2 + 4n + 4)
    Combine like terms:
    2n2+4n+42n^2 + 4n + 4

  4. Factorize:
    Factor out 22:
    2(n2+2n+2)2(n^2 + 2n + 2)

  5. Divisibility analysis:

    • The expression is clearly divisible by 2.
    • Test for higher divisibility by substituting values for nn:
      • If n=2n = 2: 2(22+22+2)=2(4+4+2)=210=202(2^2 + 2 \cdot 2 + 2) = 2(4 + 4 + 2) = 2 \cdot 10 = 20.
      • If n=4n = 4: 2(42+24+2)=2(16+8+2)=226=522(4^2 + 2 \cdot 4 + 2) = 2(16 + 8 + 2) = 2 \cdot 26 = 52.
      • 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:

  1. Can you prove the result for other pairs of even numbers?
  2. What happens if the numbers are consecutive odd numbers instead?
  3. Can you derive similar results for three consecutive numbers?
  4. Is there any case where the sum of squares is divisible by 16? Why or why not?
  5. How does this extend to numbers of different modular forms (e.g., 3k,3k+13k, 3k+1)?

Tip: Always simplify expressions and factorize them to reveal divisibility properties!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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