P4973 Magic Base of the “Poison Tumor”

hsl2363 is building a computer.

hsl2363 found computers very fun, so he plans to build one himself. After he finishes it, zrz_orz comes. hsl2363 tells him: “My computer uses the ‘**$X$-\* base**’. I really want to know how weak you are, so I’m asking you what these numbers become after converting to decimal.” A number written in “**$X$-\* base**” can be made of ASCII visible characters (excluding characters whose ASCII code is $\le 48$). Let an “**$X$-\* base**” string $s$ have length $n$, with digit positions numbered from right to left as $0$ to $n - 1$. The value it represents is the product over all positions: take the ASCII code of the character at position $i$, subtract the ASCII code of `0` (i.e., $48$), and multiply by $X^i$, then multiply all these terms together. That is, $$s=\prod \limits_{i=0} \limits^{n-1} (X^i\cdot (f(s_i)-f('0')))$$ where $f('x')$ denotes the ASCII code value of character `x`. Here is an example: If $X=3$ and $s=$`2363`, then $\begin{aligned} s&= f('2') \times 3^3 \times f('3') \times 3 ^ 2 \times f('6') \times 3 ^ 1 \times f('3') \times 3 ^ 0 \\ &= 2 \times 3^3 \times 3 \times 3 ^ 2 \times 6 \times 3 ^ 1 \times 3 \times 3 ^ 0\\ &= 78732 \end{aligned} $ zrz_orz thinks it is too hard, so he throws the problem to you. Now there are $Q$ numbers written in “$X$-\* base” (different numbers may have different $X$). You need to find their decimal representations. Because the answer may be very large, output the length of the answer.

The first line contains an integer $Q$. The next $Q$ lines: the $i$-th line contains an integer $X_i$ and a string $s_i$, meaning the $i$-th number is in “$X_i$-\* base”, and its representation in that base is $s_i$.

Output $Q$ lines. The $i$-th line contains one integer, the length of the $i$-th number’s decimal representation.

**【Constraints】** For $100\%$ of the testdata, $1 \le Q \le 100, 1\le X_i=|s_i|-1 \le 10^5$. ASCII visible characters (excluding characters whose ASCII code is $\le 48$) include the following characters (sorted in increasing ASCII code): ```text 123456789:;? @ABCDEFGHIJKLMNO PQRSTUVWXYZ[\]^_ `abcdefghijklmno pqrstuvwxyz{|}~ ``` **【Hint】** If you want to brute force it, it is impossible without big integers. Translated by ChatGPT 5