1) If , prove that

Not quite sure how to do that one..

2) If and , prove that

Suppose and

for some u in Z

for some v in Z

Now, how do I say that (a,b) = 1 = (a,c) ?

Thanks for any help.

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- July 7th 2008, 03:01 PMshadow_2145Divisibility
1) If , prove that

Not quite sure how to do that one..

2) If and , prove that

Suppose and

for some u in Z

for some v in Z

Now, how do I say that (a,b) = 1 = (a,c) ?

Thanks for any help. - July 7th 2008, 03:13 PMo_O
#1

Let (gcd of any two integers is always greater than or equal to 1).

By definition, there are some integers such that and .

In other words, dx is a common divisor of and . Since (a,b) = d, then .

However, recall that . So, - July 7th 2008, 03:51 PMJhevon
i really like o_O's solution to this. but here is how i was thinking of proving it. being very unimaginative, i like to go by definitions a lot and leave the fancy stuff to geniuses like o_O (Rofl)

disclaimer (?): to avoid having to say, "for some integers..." or "where so and so are integers..." all the time (which you should do!) just assume that whenever i introduce a new variable it is an integer.

recall the properties of the gcd:

**Definition:**means that:

(1) and

(2) If there is such that and , then

also: ....this is pretty much equivalent to condition (1), we will use this a lot.

these are the properties we will use in answering both questions. i hope you are familiar with them. now lets get to it.

Proof:

Assume . then . dividing both sides by (note ) we get , so that 1 satisfies condition (1) above. now, to show it satisfies condition (2), assume there is some such that and . then and . plug these into the equation that satisfies condition (1), we find that . so that 1 satisfies condition (2), and we have

Quote:

2) If and , prove that

Suppose and

for some u in Z

for some v in Z

Now, how do I say that (a,b) = 1 = (a,c) ?

Thanks for any help.

Proof:

Assume and . then we have:

...................(eq 1)

................(eq 2)

From (eq 1), ..............(eq 3), and .................(eq 4).

plugging in (eq 3) into (eq 2) and simplifying, we get: , so that 1 satisfies condition (1) in regards to (a,c)

plugging in (eq 4) into (eq 2) and simplifying, we get: , so that 1 satisfies condition (1) in regards to (a,b)

now, proving condition (2) for both of these is pretty much identical to the way i did it in the last question. i leave it to you. - July 7th 2008, 03:55 PMo_O
#2

Let which implies Through your reasoning, we see that . ( is a common divisor of b and c)

Similarly, if we let (implying ), we see that . ( is also a common divisor of b and c)

However, .

Edit: Ah beaten by Jhevon ;) - July 7th 2008, 04:05 PMshadow_2145
Woo, thanks. I was having problems incorporating the definitions :(

- July 7th 2008, 04:06 PMJhevon
- July 7th 2008, 04:08 PMJhevon
- July 7th 2008, 08:16 PMshadow_2145
Yes, I am okay for now. :) Thank you.