Which of the following numbers will exactly divide (49¹⁵ − 1)?

Problem Restatement

Test the divisibility of 49¹⁵ − 1 by the given options using number-theory facts.

Concept / Approach

• For any integer a, aⁿ − 1 is always divisible by (a − 1).
• If n is even, aⁿ − 1 is also divisible by (a + 1); here n = 15 (odd), so this does not apply.

Step-by-Step Reasoning

a = 49, n = 15 → a − 1 = 48Thus 49¹⁵ − 1 is divisible by 48.Check a + 1 = 50: 49 ≡ −1 (mod 50) → 49¹⁵ − 1 ≡ (−1)¹⁵ − 1 = −2 ≠ 0 (mod 50)

Final Answer

48