1. With square tiles of a side measuring an exact number of units, a room with an area of 18144 square units has been tiled in the following way: on the first day, one tile was placed, on the second day, two tiles, on the third day, three, etc. How many tiles were needed?
Solution. Suppose $n$ tiles were needed and their size is $k \times k$. Then $n k^{2}=18144=2^{5} \times 3^{4} \times 7$. There are nine possible cases for $n$, namely, $2 \times 7,2^{3} \times 7,2^{5} \times 7,2 \times 3^{2} \times 7,2^{3} \times 3^{2} \times 7,2^{5} \times 3^{2} \times 7$, $2 \times 3^{4} \times 7,2^{3} \times 3^{4} \times 7,2^{5} \times 3^{4} \times 7$. Moreover, this number must be expressible in the form $1+2+3+\cdots+N=N(N+1) / 2$ and this is only possible in the sixth case: $2^{5} \times 3^{2} \times 7=63 \times 64 / 2=2016$. To quickly rule out the other cases, we observe that $N$ and $N+1$ are coprime. For example, if $N(N+1) / 2=2^{3} \times 7$, it would have to be $N+1=2^{4}$ and $N=7$, which is impossible, etc. Therefore, 2016 tiles were needed.