If `((1+i)/(1-i))^((m)/(2))=((1+i)/(1-i))^((n)/(3))=1, (m, ninN)` then the greatest common divisor of the least values of m and n is .............

Answer

To solve the problem, we need to analyze the given equation step by step. The equation is:

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

### Step 1: Simplify \(\frac{1+i}{1-i}\)

To simplify \(\frac{1+i}{1-i}\), we can multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1+i}{1-i} \cdot \frac{1+i}{1+i} = \frac{(1+i)(1+i)}{(1-i)(1+i)} \]

Calculating the numerator:

\[ (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \]

Calculating the denominator:

\[ (1-i)(1+i) = 1 - i^2 = 1 - (-1) = 2 \]

Thus, we have:

\[ \frac{1+i}{1-i} = \frac{2i}{2} = i \]

### Step 2: Substitute back into the equation

Now substituting back into the original equation, we get:

\[ i^{\frac{m}{2}} = 1 \quad \text{and} \quad i^{\frac{n}{3}} = 1 \]

### Step 3: Determine the conditions for \(i^k = 1\)

The powers of \(i\) cycle every 4:

\[ i^0 = 1, \quad i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \ldots \]

For \(i^{\frac{m}{2}} = 1\), \(\frac{m}{2}\) must be a multiple of 4:

\[ \frac{m}{2} = 4k \quad \text{for some integer } k \implies m = 8k \]

For \(i^{\frac{n}{3}} = 1\), \(\frac{n}{3}\) must also be a multiple of 4:

\[ \frac{n}{3} = 4j \quad \text{for some integer } j \implies n = 12j \]

### Step 4: Find the least values of \(m\) and \(n\)

The least value of \(m\) occurs when \(k = 1\):

\[ m = 8 \cdot 1 = 8 \]

The least value of \(n\) occurs when \(j = 1\):

\[ n = 12 \cdot 1 = 12 \]

### Step 5: Find the greatest common divisor (GCD)

Now, we need to find the GCD of the least values of \(m\) and \(n\):

\[ \text{GCD}(8, 12) \]

The factors of 8 are \(1, 2, 4, 8\) and the factors of 12 are \(1, 2, 3, 4, 6, 12\). The common factors are \(1, 2, 4\), so the GCD is:

\[ \text{GCD}(8, 12) = 4 \]

### Final Answer

The greatest common divisor of the least values of \(m\) and \(n\) is:

\[ \boxed{4} \]