Elementary Number Theory proof

  • #1
cbarker1
Gold Member
MHB
349
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Dear Everyone,

Here is the question:

"Prove that if divides the integers and , then divides for every pair of integers and for every pair of integers."

The attempted work:

Suppose divides and divides , where . Then, and , where (Here is where I am stuck).

Do I solve for ?

If I do solve for , then it yields the
. Then, . So . Then divides .

So is it right to the proof this way?
 
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  • #2
Cbarker1 said:
Dear Everyone,

Here is the question:

"Prove that if divides the integers and , then divides for every pair of integers and for every pair of integers."

The attempted work:

Suppose divides and divides , where . Then, and , where (Here is where I am stuck).

Do I solve for ?

If I do solve for , then it yields the
. Then, . So . Then divides .

So is it right to the proof this way?
Hi Cbarker1,

and are arbitrary integers in the question, you should not use them as you do in the proof, where they are fixed integers that depend on , , and .

You can say that there are integers and such that and . Now, you have:

As is an integer, this shows that divides .
 

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