Explanation

Modulo 4: 1^5+2^5+3^5+4^5+5^5 = 1+0+243+0+1 = 1+0+3+0+1 = 5, which is congruent to 1 mod 4. The remainder is 1 - option (B). The even terms 2^5 and 4^5 are multiples of 4 and contribute 0, while 1^5, 3^5 and 5^5 leave remainders 1, 3 and 1 - summing to 5, i.e. remainder 1.