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opus
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In a book I'm reading, I'm told to prove that: If m and n are odd, then (m)(n) is odd.
The proof goes as such:
Let m=(2a+1) and n=(2b+1)
Then,
mn= (2a+1)(2b+1) = 4ab+2a+2b+1 = 2(2ab+a+b)+1 = 2t+1 ; where t= 2ab+a+b
Two questions:
When we take an expression, and assign it to a single variable, what rules must we abide by? Can we do this for any situation? My reasoning tells me the step is correct because the definition for an odd number is 2t+1, with t being any natural number. However, how can we be sure that the expression, for which we defined as t, will give a natural number? What's to say that expression can't give a negative number, which would make the solution to the proof false?
My second question is in regards to the definition of a prime as it relates to this proof. To my understanding, there are two ways to define a prime number.
1) 2m-1, where m is any integer
2) 2n+1, where n is any natural number
Would both of these definitions work for this proof?
The proof goes as such:
Let m=(2a+1) and n=(2b+1)
Then,
mn= (2a+1)(2b+1) = 4ab+2a+2b+1 = 2(2ab+a+b)+1 = 2t+1 ; where t= 2ab+a+b
Two questions:
When we take an expression, and assign it to a single variable, what rules must we abide by? Can we do this for any situation? My reasoning tells me the step is correct because the definition for an odd number is 2t+1, with t being any natural number. However, how can we be sure that the expression, for which we defined as t, will give a natural number? What's to say that expression can't give a negative number, which would make the solution to the proof false?
My second question is in regards to the definition of a prime as it relates to this proof. To my understanding, there are two ways to define a prime number.
1) 2m-1, where m is any integer
2) 2n+1, where n is any natural number
Would both of these definitions work for this proof?