ABC26GN0383 · Number System

$168=2^{3} \times 3 \times 7$ and $\underline{\mathrm{Z}}=7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=7 \times 5 \times 3^{2} \times 2^{4}=168 \times 30$ Hence, $\underline{\mathrm{Z}}$ and all the factorials greater than $\underline{\mathrm{Z}}$ are divisible by 168. Now, $N=\lfloor 1+\lfloor 2+\rfloor 3+\ldots \ldots \ldots . .+\lfloor 99+\rfloor 100$ $=\lfloor 1+\lfloor 2+\rfloor 3+\lfloor 4+\lfloor 5+\lfloor 6+$ a multiple of 168. So, the remainder obtained on dividing $N$ by 168 is the same as that obtained on dividing ( $1+12+\underline{3}+\underline{4}+\underline{5}+\underline{6}$ ) by 168. Now, $\lfloor 1+\lfloor 2+\rfloor 3+\lfloor 4+\lfloor 5+\lfloor 6=1+2+6+24+120+720$ $=873=(168 \times 5)+33$. Hence, the required remainder is 33 .