How can I prove this using the remainder theorem?

In summary, the conversation discusses proving that q^2 is exactly divisible by p+q if p^2 is also exactly divisible by p+q. The person asking the question shares their understanding of using the remainder theorem to prove this, while another person points out a flaw in their argument and provides a corrected equation. They also discuss the possibility of a calculation error.
  • #1
Hyperreality
202
0
If p^2 is exactly divisible by p+q, then proof q^2 is exactly divisible by p+q.

How do I proof this, and how do I apply the remainder theorem?

I know if f(x) = x^2 + 2x + 1, since f(-1) = 0 there fore (x + 1) is a factor of f(x).

So in this case
p^2 = p x p or p^2 x 1...
 
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  • #2
Since you specifically mention x2+ 2x+ 1, haven't you looked at (p+q)<sup>2</sup>= p<sup>2</sup>+ 2pq+ q<sup>2</sup>?
 
  • #3
This is what I've done afterwards, but I'm not sure if it is right.

Let p^2 = x^2 + 2xy + y^2.

where y^2 = y.y or -y.-y or x^2 = x.x or -x.-x.

When y = -x, or x = -y, p^2 = 0

Therefore (x + y) is a factor of p^2, also (x + y) = (p + q)

Therefore p^2 = p^2 + 2pq + p^2. p^2 = 0 when p = -q,

but (p)^2 = (-q)^2
p^2 = q^2
therefore (p+q)mod q^2 = 0

Are there any flaw in my argument or have I made any calcuation error?
 
  • #4
Sorry, it's q^2mod(p+q) = 0
 
  • #5
Originally posted by Hyperreality
This is what I've done afterwards, but I'm not sure if it is right.

Let p^2 = x^2 + 2xy + y^2.

where y^2 = y.y or -y.-y or x^2 = x.x or -x.-x.

When y = -x, or x = -y, p^2 = 0

Therefore (x + y) is a factor of p^2, also (x + y) = (p + q)

Therefore p^2 = p^2 + 2pq + p^2. p^2 = 0 when p = -q,

but (p)^2 = (-q)^2
p^2 = q^2
therefore (p+q)mod q^2 = 0

Are there any flaw in my argument or have I made any calcuation error?
p^2=P^2+2pq+q^2
-2pq=q^2
-2p=q
p=-q/2
not p=-q
 

1. What is remainder theory proof?

Remainder theory proof is a method of mathematical proof that involves using the remainder of a number when divided by another number. It is used to prove theorems and equations related to the remainders of numbers.

2. How is remainder theory proof used in mathematics?

Remainder theory proof is often used in number theory and algebra, where it is used to prove theorems related to divisibility and remainders. It can also be used in other areas of mathematics, such as geometry and calculus, to prove equations and formulas.

3. What are the steps involved in a remainder theory proof?

The steps involved in a remainder theory proof may vary depending on the specific problem, but generally they involve using the properties of remainders and modular arithmetic to show that a statement or equation is true. This may include making assumptions, using logical reasoning, and performing calculations with remainders.

4. Can remainder theory proof be used to prove all mathematical statements?

No, remainder theory proof is just one method of mathematical proof and may not be applicable to all statements or problems. Other methods, such as induction or contradiction, may be more appropriate for certain types of statements.

5. How can I improve my skills in using remainder theory proof?

To improve your skills in using remainder theory proof, you can practice solving problems and theorems related to remainders and modular arithmetic. You can also study and learn about different techniques and strategies for using remainder theory proof, and seek guidance from a mathematics teacher or tutor.

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