P5572 [CmdOI2019] Simple Number Theory Problem

Given $n,m$, find the value of the following expression: $$\sum\limits_{i=1}^n\sum\limits_{j=1}^m \varphi\left(\dfrac{{\rm lcm}(i,j)}{\gcd(i,j)}\right) \bmod 23333$$

The first line contains an integer $T$, which denotes the number of queries. The next $T$ lines each contain two integers $n,m$, representing one query.

For each query, output one integer per line, which is the answer.

For all test cases, $T\leq 3\times 10^4,\ m\leq n\leq 5\times 10^4$. | Test Point ID | $n,m\leq$ | $T$ | Time Limit | Special Property | | :--: | :--: | :--: | :--: | :--: | | #1~2 | $100$ | $100$ | $\texttt{1s}$ | | | #3~4 | $2000$ | $3\times 10^4$ | $\texttt{1s}$ | | | #5~6 | $3\times 10^4$ | $5000$ | $\texttt{2s}$ | | | #7~8 | $5\times 10^4$ | $3\times 10^4$ | $\texttt{2s}$ | $n=m$ | | #9~10 | $5\times 10^4$ | $3\times 10^4$ | $\texttt{2s}$ | | Translated by ChatGPT 5