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Suppose that z'=a+ib, i=√-1, and z=5+3i, zz'=3-4i.
(a+ib)(5+3i)=3-4i → (5a-3b) + i(3a+5b) = 3 - i4.
Equate real and imaginary parts, 5a-3b=3 and 3a+5b=-4.
Solving this you get a=3/34, b=-29/34.
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