Can Integer Solutions to 12m + 15n = 1 Be Positive?

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In summary, the author is trying to get across the idea of there being integers m and n such that 12m+15n=1, but doesn't explain how to do the proof. If you could help the author out, that would be much appreciated.
  • #1
physicsgirlie26
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Prove that:

d) there do not exsist integers m abd n such that 12m+15n=1

f) if there exist integers m and n such that 12m+15n=1, then m and n are both positive.

so far for d i have

d) since 12m is always a multiple of 3
and since 15n is always a multiple of 3, then
adding or subtracting two multiples of 3 always yields another multiple of 3, and so
12m + 15n can never equal 1 (it can only equal multiples of 3.)

but I have no idea how to do f. In the back of the book it has a hint that says, "See the statement of part (d). Can you prove that m and n are both negative whenever the antecedent is true?" I don't understand it. Can you please help me?

Thanks for the help!
 
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  • #2
d) for any integers x and y, there exists integer m and n such that xm + yn =1 if and only if gcd(x,y) = 1. Does gcd(12,15)=1?
 
  • #3
no the gcd(12, 15)=3.
 
  • #4
I don't see what the question is getting at either. If you write it as P->Q, then you've shown there are no integers m and n that satisfy the antecedent. So formally the implication is true, but you could equally well claim if m and n satisfy 12m+15n=1 then m and n are bananas.
 
  • #5
I really don't understand it either and it is killing me. -.- I have a headache because I'm trying to figure it out and I can't. haha Does anyone have any other suggestions on how to go about this proof?
 
  • #6
That's all the question is trying to get at... in a statement P->Q, if the P part is false, the entire statement is true by default. This is chapter 1, don't over think it too much. Just like Dick said "If there exist integers m and n such that 12m+15n=1, then m and n are bananas" is also a very true statement. The author is just trying to get a very simple concept across, in a fairly convoluted way.
 
  • #7
But how would you prove that that m and n are both negative whenever the antecedent is true?
 
  • #8
The thing is... the antecedent isn't true. So if you were asked to prove that "If there exists integers m and n such that 12m+15n=1 then m and n are negative" all you'd do is prove that the antecedent is false and say "by default this statement is true."...
 
  • #9
Ok. Thanks for all of your help! :0)
 

Related to Can Integer Solutions to 12m + 15n = 1 Be Positive?

1. What is a mathematical proof?

A mathematical proof is a logical argument that demonstrates the validity of a mathematical statement or theorem. It is a step-by-step process that uses previously established mathematical concepts and rules to show that a statement is true.

2. Why is it important to write a proof?

Writing a proof helps to develop critical thinking and problem-solving skills. It also helps to solidify understanding of mathematical concepts and serves as a universal language for communicating ideas and theories in the field of mathematics.

3. How do I start writing a proof?

The first step in writing a proof is to clearly define the statement or theorem you are trying to prove. Then, identify any given information, and use previously established mathematical concepts and rules to logically connect these pieces of information to the statement you are trying to prove.

4. How can I check if my proof is correct?

One way to check the correctness of a proof is to go through each step and make sure that it follows logically from the previous steps. You can also have someone else review your proof and provide feedback or try to find flaws in your argument.

5. Are there any specific tips for writing a proof?

Some tips for writing a proof include being organized and clear, using precise mathematical language, and providing explanations for each step. It is also important to check for any logical fallacies and to revise and refine your proof until it is clear and concise.

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