If a prime number on division by 4 gives a remainder of 1, then that number can be expressed as: A. sum of squares of two natural numbers B. sum of cubes of two natural numbers C. sum of square roots of two natural numbers D. sum of cube roots of two natural numbers

GATE 2012 · General Aptitude · Number Theory · medium

Answer: A prime p ≡ 1 (mod 4) can be expressed as the sum of squares of two natural numbers (Fermat's two-square theorem). Correct answer: A. Note: question excluded from GATE 2012 AE scoring (answer key X).

1. State Fermat's theorem on sums of two squares: Given p ≡ 1 (mod 4), Fermat's theorem guarantees a representation p = a^2 + b^2.
2. Verify with examples: All primes congruent to 1 mod 4 can be expressed as sums of two squares of natural numbers.
3. Note the exclusion: The GATE 2012 AE official answer key records this as X (excluded). The correct mathematical answer is A: sum of squares of two natural numbers.