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 is an integer, and is divisible by 8, then is divisible by 8," we need to find a counterexample where is divisible by 8 but is not divisible by 8.
Counterexample:
Let .
-
.
is divisible by (since ). -
is not divisible by 8 (since is not an integer).
Thus, satisfies the conditions of the hypothesis ( divisible by ) but violates the conclusion ( divisible by ). 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:
- Why does being divisible by 8 not necessarily imply is divisible by 8?
- Can this logic apply to divisibility by numbers other than 8?
- What are the general rules for divisibility of squares in integers?
- How can counterexamples be systematically constructed in mathematics?
- 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