The remainder obtained when (⌊ 1)2 + (⌊ 2)2 + (⌊ 3)2 + ... + (⌊ 100)2 is divided by 102 is ;

Question

The remainder obtained when (⌊ 1)2 + (⌊ 2)2 + (⌊ 3)2 + ... + (⌊ 100)2 is divided by 102 is ; (A) 14 (B) 17 (C) 28 (D) 27

Tardigrade Admin · Accepted Answer

Terms greater than 5! i.e., (5 !)2, ( 6 !)2, ..., (100!)2 is divisible by 100 ∴ For terms (5 !)2, ( 6 !)2, ..., (100!)2 remainder is 0. Now consider (1 !)2 + (2 !)2 + (3 !)2 + (4 !)2 = 1 + 4+ 36+ 576 = 617 When 617 is divided by 100, its remainder is 17. ∴ Required remainder is 17.

Q. The remainder obtained when (⌊1)2+(⌊2)2+(⌊3)2+...+(⌊100)2 is divided by 102 is ;

14

17

28

27

Terms greater than 5! i.e., (5!)2,(6!)2,...,(100!)2 is divisible by 100 ∴ For terms (5!)2,(6!)2, ..., (100!)2 remainder is 0. Now consider (1!)2+(2!)2+(3!)2+(4!)2 = 1 + 4+ 36+ 576 = 617 When 617 is divided by 100, its remainder is 17. ∴ Required remainder is 17.

Q. The remainder obtained when $(⌊ 1)^{2} + (⌊ 2)^{2} + (⌊ 3)^{2} + ... + (⌊ 100)^{2}$ is divided by $1 0^{2}$ is ;