Given that abc = -37 + 36i; b = -2 + 3i; c = 1 + 2i, what is a?

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Substituting the given values for b and c into the equation for abc,

a(-2 + 3i)(1 + 2i) = -37 + 36i

Multiplying out the terms in brackets,

a(-2 - 4i + 3i - 6) = -37 + 36i

Collecting like terms and multiplying through by -1,

a(8 + i) = 37 - 36i

The complex number a can be represented as m + ni, where m and n are constants we need to find.

(8 + i)(m + ni) = 37 - 36i

Multiplying out the terms in brackets,

8m + 8ni + mi - n = 37 - 36i

Collecting like terms and equating the real and imaginary parts, we end up with two simultaneous equations for m and n.

8m - n = 37 (from real part)

8n + m = -36 (from imaginary part)

Rearranging the first equation, we find that n = 8m - 37. Substituting this into the second equation,

8(8m - 37) + m = -36

64m - 296 + m = -36

65m - 296 = -36

65m = 260

m = 4

Subsituting this value for m back into the second equation,

8n + 4 = -36

8n = -40

n = -5

Putting it all together,

a = 4 - 5i

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