The least non-negative remainder when \(2^{75}\) is divided by 5 will be:-
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3
The key is a power of 2 that leaves remainder 1 on division by 5:
\(2^4 = 16 = 3 \times 5 + 1\)
\(2^4 \equiv 1 \pmod{5}\)
The exponent splits as a multiple of 4 plus a remainder:
\(75 = 4 \times 18 + 3\)
\(2^{75} = (2^4)^{18} \times 2^3\)
\(2^{75} \equiv 1^{18} \times 8 \pmod{5}\)
\(8 \equiv 3 \pmod{5}\)
So the remainder is 3.