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

[physicsgirlie26]

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!

 

[quasar987]

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?

 

[physicsgirlie26]

no the gcd(12, 15)=3.

 

[Dick]

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.

 

[physicsgirlie26]

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?

 

[alec_tronn]

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.

 

[physicsgirlie26]

But how would you prove that that m and n are both negative whenever the antecedent is true?

 

[alec_tronn]

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."...

 

[physicsgirlie26]

Ok. Thanks for all of your help! :0)