Question:
The remainder obtained when $(\lfloor 1)^2 + (\lfloor 2)^2 + (\lfloor 3)^2 + ... + (\lfloor 100)^2$ is divided by $10^2$ is ;
The remainder obtained when $(\lfloor 1)^2 + (\lfloor 2)^2 + (\lfloor 3)^2 + ... + (\lfloor 100)^2$ is divided by $10^2$ is ;
Updated On: May 14, 2024
- 14
- 17
- 28
- 27
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The Correct Option is B
Solution and Explanation
Terms greater than 5!
i.e., $(5 !)^2, ( 6 !)^2, ..., (100!)^2$ is divisible
by 100
$\therefore$ 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.
$\therefore$ 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.