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Let a + bi be the reciprocal. So (a + bi)(2 - 5i) = 1
(a + bi)(2 - 5i) = 2a - 5ai + 2bi + 5b = (2a + 5b) + (2b - 5a)i
Therefore 2a + 5b = 1 and 2b - 5a = 0. Solving the simultaneous equations, we find that a = 2/29 and b = 5/29.
So the reciprocal of 2 - 5i is 2/29 + 5i/29.
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