If then the greatest common divisor of the least values of m and n is ______

Engineering

Mathematics

Introduction to Complex Number

Question

If ${(\frac{1 + i}{1 - i})}^{\frac{m}{2}} - {(\frac{1 + i}{i - 1})}^{\frac{n}{3}} = 1, (m, n \in N)$ then the greatest common divisor of the least values of m and n is _______ .

Soltion to the Given Problem

\[ \left( \frac{1+i}{1-i} \right)^{\frac{m}{2}} - \left( \frac{1+i}{i-1} \right)^{\frac{n}{3}} = 1 \]

We define:

\[ z = \frac{1+i}{1-i} \]

Expressing in polar form:

\[ 1+i = \sqrt{2} e^{i\pi/4}, \quad 1-i = \sqrt{2} e^{-i\pi/4} \]

Thus,

\[ z = \frac{\sqrt{2} e^{i\pi/4}}{\sqrt{2} e^{-i\pi/4}} = e^{i\pi/2} = i \]

Similarly,

\[ \frac{1+i}{i-1} = -i \]

Rewriting the equation:

\[ i^{\frac{m}{2}} - (-i)^{\frac{n}{3}} = 1 \]

From the cyclic properties of i and -i:

The smallest m satisfying $ i^{m/2} = 1 $ is $ m = 8 $.
The smallest n satisfying $ (-i)^{n/3} = 1 $ is $ n = 12 $.

Finding GCD:

\[ \gcd(8, 12) = 4 \]

Final Answer: 4

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