Second Floor

• $\lfloor \cdot \rfloor$ denotes the floor function .

$y=\left\lfloor 2x+5 \right\rfloor =3\left\lfloor x-4 \right\rfloor \quad \quad \quad (Given)$

$\left\lfloor 2x \right\rfloor +5=3\left\lfloor x \right\rfloor -3\left( 4 \right)$

$\left\lfloor 2x \right\rfloor =3\left\lfloor x \right\rfloor -17$

$Let\quad x=I+f\quad \quad Where\quad I\quad is\quad an\quad Integer\quad and\quad f\quad is\quad a\quad Fraction\quad 0\le f<1.$

$\left\lfloor 2I+2f \right\rfloor =3\left\lfloor I+f \right\rfloor -17\\ 2I+\left\lfloor 2f \right\rfloor =3I+3\left\lfloor f \right\rfloor -17\\ 2I+\left\lfloor 2f \right\rfloor =3I-17\\ \left\lfloor 2f \right\rfloor =I-17$

$0\le f<1\\ 0\le 2f<2$

$\overbrace { \underbrace { Case1 } }$

$0\le 2f<1\quad =>\quad 0\le f<0.5\\ \left\lfloor 2f \right\rfloor =0=I-17\\ =>\quad I=17\\ =>\quad 17\le x<17.5\quad \quad \quad as\quad x=I+f\\ \left\lfloor x \right\rfloor =17\\ \\ 51\le 3x<52.5\\ \left\lfloor 3x \right\rfloor =\quad 51\quad or\quad 52\\ \\ As\quad y=3\left\lfloor x \right\rfloor -12\quad \quad \quad \quad (Given)\\ y=3(17)-12=39\\ \\ \left\lfloor 3x+y \right\rfloor =\left\lfloor 3x+39 \right\rfloor =\left\lfloor 3x \right\rfloor +39=51+39\quad or\quad 52+39\\ \\ \left\lfloor 3x+y \right\rfloor =90\quad or\quad 91\quad$

$\overbrace { \underbrace { Case2 } }$

$1\le 2f<2\quad \quad 0.5\le f<1\\ \\ \left\lfloor 2f \right\rfloor =I-17\\ \\ I=18$

$18.5\le x<19\quad \quad \left\lfloor x \right\rfloor =18\quad \quad (As\quad x=I+f)\\ 55.5\le 3x<57\\ \left\lfloor 3x \right\rfloor =\quad 55\quad or\quad 56\\ y=3\left\lfloor x \right\rfloor -12=3(18)-12=42\\ \\ \left\lfloor 3x+y \right\rfloor =\quad 55+42\quad or\quad 56+42\\ \left\lfloor 3x+y \right\rfloor =98\quad or\quad 97$

$A=\quad Product\quad of\quad All\quad Values\\ A=90*91*97*98$

$Now\quad A(mod\quad2016)=90*91*98*97\quad (mod\quad2016)\\ =8190*9506(mod\quad2016)=126*1442\quad (mod\quad2016)\\ =181692\quad (mod\quad2016)=\quad252$

I Hope you enjoyed My Solution.