Question:

The remainder obtained when (⌊1)2+(⌊2)2+(⌊3)2+...+(⌊100)2(\lfloor 1)^2 + (\lfloor 2)^2 + (\lfloor 3)^2 + ... + (\lfloor 100)^2 is divided by 10210^2 is ;

Updated On: May 14, 2024
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The Correct Option is B

Solution and Explanation

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

Binomial Theorem

The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is 

Properties of Binomial Theorem

  • The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
  • There are (n+1) terms in the expansion of (x+y)n.
  • The first and the last terms are xn and yn respectively.
  • From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
  • The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.