Math Problem Statement

Find a counterexample that disproves the statement "If n is an integer, and n2 is divisible by 8, then n is divisible by 8."

Solution

To disprove the statement "If nn is an integer, and n2n^2 is divisible by 8, then nn is divisible by 8," we need to find a counterexample where n2n^2 is divisible by 8 but nn is not divisible by 8.


Counterexample:

Let n=4n = 4.

  1. n2=42=16n^2 = 4^2 = 16.
    1616 is divisible by 88 (since 16÷8=216 \div 8 = 2).

  2. n=4n = 4 is not divisible by 8 (since 4÷84 \div 8 is not an integer).

Thus, n=4n = 4 satisfies the conditions of the hypothesis (n2n^2 divisible by 88) but violates the conclusion (nn divisible by 88). This is a counterexample that disproves the statement.


Would you like a detailed explanation of the logic or any further questions?
Here are 5 questions related to this topic:

  1. Why does n2n^2 being divisible by 8 not necessarily imply nn is divisible by 8?
  2. Can this logic apply to divisibility by numbers other than 8?
  3. What are the general rules for divisibility of squares in integers?
  4. How can counterexamples be systematically constructed in mathematics?
  5. How does this relate to the concept of necessary and sufficient conditions?

Tip: When disproving a statement, always seek a specific example that satisfies the hypothesis but fails the conclusion!

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Math Problem Analysis

Mathematical Concepts

Number Theory
Divisibility
Counterexamples

Formulas

n^2 divisible by k implies specific properties of n

Theorems

Basic properties of divisibility
Implications and counterexamples in logic

Suitable Grade Level

Grades 10-12