Explanation:

To determine the largest number that divides both \(2^{35} - 1\) and \(2^{91} - 1\), we must examine the common divisors.

- The divisors of \(2^n - 1\) are also divisors of \(2^{km} - 1\) for all \(m\), where \(k\) is a divisor of \(n\).

- The greatest common divisor of \(2^{35} - 1\) and \(2^{91} - 1\) is \(2^{\text{gcd}(35, 91)} - 1\).

- The gcd of 35 and 91 is 7, as both share this factor.

- Therefore, \(2^7 - 1 = 127\) is the largest number that divides both.

- Option 1: 34 - Incorrect, since \(34\) is not a factor of \(2^7 - 1 = 127\).

- Option 2: 90 - Incorrect, for similar reasons as option 1.

- Option 3: 127 - Correct, matches \(2^7 - 1\).

- Option 4: 129 - Incorrect, exceeds the largest divisor \(127\).

Option 3: 127 is the correct answer.