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

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Q. The remainder obtained when $(\lfloor 1)^2 + (\lfloor 2)^2 + (\lfloor 3)^2 + ... + (\lfloor 100)^2$ is divided by $10^2$ is ;

KCETKCET 2006Permutations and Combinations

Solution:

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.