1985 AIME Problems/Problem 3

Problem

Find $c$ if $a$ , $b$ , and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$ , where $i^2 = -1$ .

Solution

Expanding out both sides of the given equation we have $c + 107i = (a^3 - 3ab^2) + (3a^2b - b^3)i$ . Two complex numbers are equal if and only if their real parts and imaginary parts are equal, so $c = a^3 - 3ab^2$ and $107 = 3a^2b - b^3 = (3a^2 - b^2)b$ . Since $a, b$ are integers, this means $b$ is a divisor of 107, which is a prime number. Thus either $b = 1$ or $b = 107$ . If $b = 107$ , $3a^2 - 107^2 = 1$ so $3a^2 = 107^2 + 1$ , but $107^2 + 1$ is not divisible by 3, a contradiction. Thus we must have $b = 1$ , $3a^2 = 108$ so $a^2 = 36$ and $a = 6$ (since we know $a$ is positive). Thus $c = 6^3 - 3\cdot 6 = 198$ .

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1985 AIME Problems/Problem 3

Problem

Solution

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