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The greatest prime factor of (3^10 - 3^4)^50 is A. 13 B. 17 C. 3 D. 11

GATE 2024 · General Aptitude · Algebra · medium

Answer: The greatest prime factor is 13 (Option A).

  1. Drop the 50th power — it does not change prime factors: Greatest prime factor of (3^10 - 3^4)^50 = greatest prime factor of (3^10 - 3^4).
  2. Factor out the common power 3^4: 3^10 - 3^4 = 3^4 * (3^6 - 1) = 81 * (729 - 1) = 81 * 728
  3. Factorize 728: 728 / 2 = 364; 364 / 2 = 182; 182 / 2 = 91; 91 = 7 * 13. So 728 = 2^3 * 7 * 13.
  4. Identify the greatest prime: All prime factors: 2 (from 728), 3 (from 3^4), 7 (from 728), 13 (from 728). Greatest = 13.