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

Solution

Maths sir · 4 steps

That fraction is just i, so each power must be a multiple of 4 for the answer to be 1.

1. Simplify the base. Multiply top and bottom by 1 + i: (1+i)²/(1-i)(1+i) = 2i/2 = i. Both conditions become powers of i equal to 1.

   i^(m/2) = 1 and i^(n/3) = 1

2. Powers of i. The powers of i repeat in a cycle of 4: i, -1, -i, 1. So iᵏ = 1 only when k is a multiple of 4.
3. Least m and n. m/2 = 4 gives the least m = 8. n/3 = 4 gives the least n = 12.
4. So. gcd(8, 12) = 4, so the answer is 4.

The common slip: 24. Taking the LCM of 8 and 12 instead of the GCD gives 24, not 4.

The idea to keep. Whenever you see (1+i)/(1-i), write i straight away. Then iᵏ = 1 needs k to be a multiple of 4, because the powers of i repeat every 4.

Answer from the NTA final answer key. Solution by PrepWiser.