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Question

Prove that, if a, b, c and d be positive rationals such that, a+b=c+d, then either a=c and b=d or b and d are squares of rationals.

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Solution

If a=c, then a+b=c+db=db=d

So, let ac. Then, there exists a positive rational number x such that a=c+x.

Now,

a+b=c+d

c+x+b=c+d [a=c+x]

x+b=d

(x+b)2=(d)2

x2+2bx+b=d

b=dx2b2x

b is rational [d,x,b are rationalsdx2b22xis rational]

b is the square of a rational number.

From(i), we have

d=x+b

d is rational

d is the square of a rational number.

Hence, either a=c and b=d or b and d are the squares of rationals.

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