Proofs Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Prove: The product of two odd integers is odd.

Solution

  1. 1
    Let a=2m+1a = 2m+1 and b=2n+1b = 2n+1 for integers m,nm, n.
  2. 2
    Then ab=(2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1ab = (2m+1)(2n+1) = 4mn + 2m + 2n + 1 = 2(2mn+m+n) + 1.
  3. 3
    Since 2mn+m+n2mn+m+n is an integer, abab is odd.

Answer

ab=2(2mn+m+n)+1 is odd.ab = 2(2mn+m+n)+1 \text{ is odd.}
Expanding the product and rewriting in the form 2k+12k+1 proves the result directly.

About Proofs

A mathematical proof is a rigorous logical argument that demonstrates the truth of a statement beyond doubt, proceeding from accepted axioms and previously proven results through valid inference rules.

Learn more about Proofs →

More Proofs Examples

Example 1 easy

Prove directly: The sum of two even integers is even.

Example 2 medium

Prove directly: For any integer [formula], if [formula] is odd then [formula] is odd.

Example 4 medium

Prove directly: if [formula] is even, then [formula] is odd.

← All Proofs Examples Proofs Concept