m is odd)

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Mathematical Proofs

Mathematical Proofs How to Write a Proof Synthesizing definitions intuitions and conventions Proofs on Numbers Working with odd and even numbers Universal and Existential Statements Two important classes of statements Proofs on Sets From Venn diagrams to rigorous math What is a Proof?

Which statement is sufficient if M is not odd?
1. If we take m=10, the result is 5 which is not even. Therefore statement is sufficient in answering No m is not odd. 2. 5-3=2 which is even. Therefore statement 2 is sufficient because it also has a unique Yes answer that m is odd. Can someone please help me understand this. Is my approach correct? If m is an integer, is m odd?
What if M is odd?
Assume m is odd. By the Division Algorithm, there exist integers q and r, with 0 ≤ r < 2 such that m = 2 q + r. There are only two non-negative integers less than 2. They are 0 and 1. If r = 0, then m = 2 q, so m is even, contrary to our hypothesis that m is odd. Hence, r = 1. Therefore, m = 2 q + 1. Assume there exists q ∈ Z such that m = 2 q + 1.
What does m/2 is not an even integer mean?
Only integers can be even or odd, so "m/2 is not an even integer" means that m/2 is odd (integer) or not an integer at all. I have just a quick question. How can you use 10 for an example in your explaination of m/2. m must always be a negative number. So actually 1 is not sufficient because you would never get a integer. Right?
Is m an odd integer or a non-integer?
Therefore, if m/2 is not an even integer, then m=4 is excluded from the list of possibilities. This leaves m=1, m=2 and m=3. M/2 can therefore be an odd integer or a non-integer. Since we have examples of both, we cannot conclude with certainty whether m is an odd integer, it can be either 1 or 2 or 3 (or 5 or 6 or 7...)

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