Exercise · ZW-DIRECT-PROOF-009 · ★★★ Advanced
To prove that if \(n\) is odd then \(8\mid n^2-1\), write \(n=2k+1\). What is \(n^2-1\) equal to?
Choose one option.
Show answer
- A. \(4k(k+1)\), a multiple of 8 because \(k(k+1)\) is even
Explanation
\((2k+1)^2-1=4k^2+4k=4k(k+1)\). Since \(k(k+1)\) is a product of two consecutive integers, it is even, so \(4k(k+1)\) is a multiple of 8.
Review the concept: Direct proof
A direct proof of an implication \(p\Rightarrow q\) starts from the hypothesis \(p\) and, using definitions and previously proved results, chains steps until it reaches the conclusion \(q\).
\[p\Rightarrow\cdots\Rightarrow q\]
More practice problems: direct proof
- ZW-DIRECT-PROOF-001 To prove directly that if n is even then n^2 is even, one starts by writing:
- ZW-DIRECT-PROOF-002 If n=2k+1, what is n^2 equal to?
- ZW-DIRECT-PROOF-003 Let a=2k+1 and b=2j+1 be odd. Which expression shows that a+b is even?
- ZW-DIRECT-PROOF-004 In a direct proof of p⇒ q, what do we do?
- ZW-DIRECT-PROOF-005 Which of these starts is a direct proof of «if n is odd then n+1 is even»?
- ZW-DIRECT-PROOF-006 Let a=2x and b=2y be even. What follows about ab?
- ZW-DIRECT-PROOF-007 We know a| b and b| c, i.e. b=ak and c=bm. How do we conclude a| c?
- ZW-DIRECT-PROOF-008 Does checking that 3^2=9, 5^2=25 and 7^2=49 are odd prove that the square of every odd number is od…