What is the remainder when 53¹⁴⁷ − 27¹⁴⁷ is divided by 26?

Question

What is the remainder when 53¹⁴⁷ − 27¹⁴⁷ is divided by 26?

1. A. 0 (Correct answer)
2. B. 25
3. C. 1
4. D. 2

Correct Answer

Option A — 0

Detailed Solution & Explanation

The correct answer is 0.

Key Points

• There is a standard divisibility result: aⁿ − bⁿ is always divisible by (a − b) for every positive integer n.
• Here a = 53 and b = 27, so a − b = 26.
• Since 53¹⁴⁷ − 27¹⁴⁷ is divisible by 26, the remainder on division by 26 is 0.

Additional Information

• Companion results: aⁿ − bⁿ is divisible by (a + b) when n is even, and aⁿ + bⁿ is divisible by (a + b) when n is odd.
• The exponent 147 is a red herring — the factor (a − b) divides the difference for any n, so its value never matters.

*Ministry of Papers worked solution. The Commission has not yet published the official key for this paper; this answer will be reconciled with it on release.*