Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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Agnishom
2013-03-29 18:58:11

And you can use any numbers to prove that, such as Nehushtan's example?

Yes, as long as it satisfies the condition. Remember that just one counterexample is ok to disprove anything

bob bundy
2013-03-29 08:49:30

Q1 is no

and yes, that's a good example.

If p1, p2, and p3  are all different primes then you can make a suitable example like this

(p1 x p1) | (p1 x p2) x ( p1 x p3) but not (p1 x p1) | (p1 x p2) and  not (p1 x p1) | (p1 x p3)

Bob

rhymin
2013-03-29 04:55:42

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.

One more question:

So for problem #1, the answer is no.

And you can use any numbers to prove that, such as Nehushtan's example?

9|6 x 15 but 9 is not a factor of 6 or 15   Would that be a good way to explain that it is false?

bob bundy
2013-03-28 21:40:25

hi rhymin

Sorry about the earlier muddle over the | symbol.  This next bit is my excuse.  Skip it if you like but it makes me feel better to have an excuse.

EXCUSE.  It must be 45 years since I last saw that symbol in some number theory at Uni.  What I should have done is looked it up rather than relying on a faulty memory.  But I think the symbol is poorly chosen.

REASON.  If we want to put "42 divided by 6" into symbols we can say 42 ÷ 6   or 42/6 or even

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.

Bob

bobbym
2013-03-28 18:28:41

Hi;

Who is the author?

The linear diophantine equation ax+by = c where c = (a,b) is computed using the extended GCD algorithm or some other and then Bezouts identity is used to get them all.

rhymin
2013-03-28 14:15:57

#### rhymin wrote:

So for problem #1, the answer is no.

And you can use any numbers to prove that, such as Nehushtan's example?

9|6 x 15 but 9 is not a factor of 6 or 15   Would that be a good way to explain that it is false?

BTW, I just wanted to confirm that this is correct?

rhymin
2013-03-28 14:08:54

The textbook is Discrete and Combinational Mathematics.

Ahh, i see that Euclid's algorithm is a way to find the GCD.  Awesome.

Could you please shine some light on the second part of the problem?

bobbym
2013-03-28 10:23:21

What textbook are you working out of?

You can compute it or you can use a program or you can use a site. Which do you prefer?

Sorry, but I just don't get how 8a and -63b were calculated.

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.

rhymin
2013-03-28 10:21:13

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.

Sorry, but I just don't get how 8a and -63b were calculated.

I looked up Linear Relationship - A statistical term used to describe the relationship between a variable and a constant. Linear relationships can be expressed in a graphical format where the variable and the constant are connected via a straight line or in a mathematical format where the independent variable is multiplied by the slope coefficient, added by a constant, which determines the dependent variable.

But that is just adding to the confusion a bit.  Sorry, I know this can be frustrating as I'm not very smart in this area.

bobbym
2013-03-28 10:11:49

Hi;

a and b are given and t and s were calculated.

The question asks for a linear relationship between a,b and gcd(a,b).

rhymin
2013-03-28 06:49:55

#### bobbym wrote:

Hi;

2)

I am not sure I understand the question.

a = 1820
b = 231

(1820,231) = 7

1820 t +231 s = 7

t = 8
s = -63

8a - 63b = 7

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?

I'm not sure where 8a or -63b comes from either.

Any explanation is highly appreciated!

rhymin
2013-03-28 06:46:58

So for problem #1, the answer is no.

And you can use any numbers to prove that, such as Nehushtan's example?

9|6 x 15 but 9 is not a factor of 6 or 15   Would that be a good way to explain that it is false?

bob bundy
2013-03-28 05:13:21

My apologies rhymin.

Nehushtan and Agnishom are right and I am wrong about what that symbol means.

Now you have three examples that show the original statement is false.

Bob

rhymin
2013-03-28 04:15:37

And thank you everyone for your replies!

rhymin
2013-03-28 04:14:16