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Ramanujan and Hardy played a game where they both picked a complex number. If the product of their numbers was 32 - 8i, and Hardy picked 5 + 3i, what number did Ramanujan pick?

 Jul 2, 2018
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Simplify the following:
(-8 i + 32)/(3 i + 5)

Factor 8 out of 32 - 8 i giving 8 (4 - i):
(8 (-i + 4))/(3 i + 5)

Multiply numerator and denominator of (8 (-i + 4))/(3 i + 5) by 5 - 3 i:
(8 (-i + 4) (-3 i + 5))/((3 i + 5) (-3 i + 5))

(5 + 3 i) (5 - 3 i) = 5×5 + 5 (-3 i) + 3 i×5 + 3 i (-3 i) = 25 - 15 i + 15 i + 9 = 34:
(8 (-i + 4) (-3 i + 5))/34

The gcd of 8 and 34 is 2, so (8 (-i + 4) (-3 i + 5))/34 = ((2×4) (-i + 4) (-3 i + 5))/(2×17) = 2/2×(4 (-i + 4) (-3 i + 5))/17 = (4 (-i + 4) (-3 i + 5))/17:
(4 (-i + 4) (-3 i + 5))/17

(4 - i) (5 - 3 i) = 4×5 + 4 (-3 i) - i×5 - i (-3 i) = 20 - 12 i - 5 i - 3 = 17 - 17 i:
(4 -17 i + 17)/17

Factor 17 out of 17 - 17 i giving 17 (1 - i):
(4×17 (-i + 1))/17
Combine powers. (4×17 (-i + 1))/17 = 4×17^(1 - 1) (-i + 1):
4×17^(1 - 1) (-i + 1)

1 - 1 = 0:
4×17^0 (-i + 1)

17^0 = 1:
4 (-i + 1) = 4 - 4i - Ramanujan's number.

 Jul 2, 2018

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