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Distance Formula Using Complex Numbers

If a and ...

Question

If $a$ and $b$ are positive integer such that $N = (a + i b)^{3} - 107 i$ is a positive integer then find the value of $\frac{N}{2}$ .

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Solution

$N = (a + i b)^{3} - 107 i = (a^{3} - 3 a b^{2}) + (- b^{3} + 3 a^{2} b - 107) i$
N is possible integer $\Rightarrow 3 a^{2} b - b^{3} = 107$
$a = 6, b = 1$ (trial & error solution)
$N = a^{3} - 3 a b^{2} = 6^{3} - 3.6, 216 - 18 = 198 N / 2 = 99$

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Distance Formula Using Complex Numbers

Standard XII Mathematics