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Yes, as long as it satisfies the condition. Remember that just one counterexample is ok to disprove anything
Q1 is no
Bob Bundy, no worries at all. The confusion actually helped me more because I don't think I will ever forget it now because of this discussion. Also, thank you so much for showing how you solved for that.
Now, division must have come long before the concept of "is it divible by" so someone must have made up that definition. Now he/she could have defined " divides " or " is divisible by ". The result is mathematically equivalent; it's all in the way you express the property. So why, oh why, did this person choose " divides " which puts the first number second and the second number first and just to be really confusing invent the symbol | for it, when | is already heavily used in maths to mean other things, and looks a lot like the symbol for divided by \ ???
No wonder I got mixed up.
END OF EXCUSE.
So, to make amends here is a method for creating the linear combination that doesn't require a computer.
We want integers s and t so that
Divide the larger number by the smaller ( 260/33)
Divide the larger number by the smaller (33/29)
Divide the larger number by the smaller (29/4)
When one of the 'coefficients' is 1 you can stop this process and jump to simultaneous equations.
Solving gives t = -25 and s = 197.
This pair are different from bobbym's pair but both sets of answers for s and t work. There are, in fact, an infinite number of solutions so best wishes to any teacher who has to check them all.
BTW, I just wanted to confirm that this is correct?
The textbook is Discrete and Combinational Mathematics.
What textbook are you working out of?
I never waste time using a hand method to do what a machine can do better unless asked. First the GCD and then I will show how to calculate it.
What is the easiest way to figure out the GCD of 2 large numbers besides manually trying to divide by random numbers? I see now 7 is the GCD.
I'm still confused by this. Are t and s just random letters picked to help solve the problem? I also don't know where the 7 comes from unless you divide 1820 by 231 and leaving the remainder out. Is this called the quotient?
So for problem #1, the answer is no.
My apologies rhymin.
And thank you everyone for your replies!
Bobby, sorry about that.