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That's simply fastastix and brilliant. Thank you for a million times. Your help is really too grateful. Thank you.
I appreciate your help, and in fact I have tried expanding and factorizing that for hours, but I just wasn't successful. I tried using (ac+bd)² + 4(ac+bd)(ad+bc) + (ad+bc)², but there's 12abcd missing, as you expand you can see, and that was quite close indeed.
Also I tried:
1. Trigonometry. Substitute sin and cos, sin² + cos² is 1, so maybe this make the expansion simpler? But then I realize that the 'a' and 'b' wouldn't be integers if it's sin/cos, this method is not possible.
2. Matrix. (a) (1 2) is actually, and magically, a²+4ab+b². But then it would be quadratic form and I don't know how to carry on with this as I haven't learnt it.
3. Number theory and some logic. I thought, since x, y are in the set A, so x,y must also be integers. Then I tried substituting x,y as a,b, and performed so transformation, but there's too little for me to carry on with.
This is a piece of work that I needed to hand in before the end of this Friday. Any help or ideas to solve this problem would be much appreciated.
b 2 1
Let A be a set of integers that can be expressed as a²+4ab+b², where a,b are integers.
If x,y are in A, then prove that xy is also in A.
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