2009 AIME I Problems/Problem 2

Problem

There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that

$\frac {z}{z + n} = 4i.$

Find $n$ .

1st Solution

Let $z = a + 164i$ .

Then $\frac {a + 164i}{a + 164i + n} = 4i$ and $a + 164i = \left (4i \right ) \left (a + n + 164i \right ) = 4i \left (a + n \right ) - 656.$

By comparing coefficients, equating the real terms on the leftmost and rightmost side of the equation,

we conclude that $a = -656.$

By equating the imaginary terms on each side of the equation,

we conclude that $164i = 4i \left (a + n \right ) = 4i \left (-656 + n \right ).$

We now have an equation for $n$ : $4i \left (-656 + n \right ) = 164i,$

and this equation shows that $n = \boxed{697}.$

2nd Solution

$\frac {z}{z+n}=4i$

$1-\frac {n}{z+n}=4i$

$1-4i=\frac {n}{z+n}$

$\frac {1}{1-4i}=\frac {z+n}{n}$

$\frac {1+4i}{17}=\frac {z}{n}+1$

Since their imaginary part has to be equal,

$\frac {4i}{17}=\frac {164i}{n}$

$n=\frac {(164)(17)}{4}=697$

$n = \boxed{697}.$

2009 AIME I (Problems • Answer Key • Resources)
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