If \(1^{100}+2^{100}+3^{100}+4^{100}+5^{100} \equiv R \pmod{5}\), then value of R is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4
Fourth powers of 1, 2, 3, 4 all leave remainder 1 on division by 5:
\(1^4 = 1\), \(2^4 = 16\), \(3^4 = 81\), \(4^4 = 256\), each \(\equiv 1 \pmod{5}\)
Since \(100 = 4 \times 25\), raising a congruence to a power gives, for a = 1, 2, 3, 4:
\(a^{100} = \left(a^4\right)^{25} \equiv 1^{25} = 1 \pmod{5}\)
And \(5^{100} \equiv 0 \pmod{5}\). Adding:
\(R \equiv 1+1+1+1+0 = 4 \pmod{5}\)