Score : $400$ points

问题描述

我们有一个正整数 $a$。另外，黑板上写有一个以 base $10$ 表示的数字。 设 $x$ 为黑板上的数字。Takahashi 可以通过以下操作来改变这个数字。

• 擦除 $x$ 并以 base $10$ 写出 $x$ 乘以 $a$ 的结果。
• 将 $x$ 视作字符串并将最右侧的数字移动到开头。
  这个操作只能在 $x \geq 10$ 且 $x$ 不是 $10$ 的倍数时执行。

例如，当 $a = 2, x = 123$ 时，Takahashi 可以执行以下任一操作：

• 擦除 $x$ 并写出 $x \times a = 123 \times 2 = 246$。
• 将 $x$ 视作字符串并把 123 的最右侧数字 3 移动到开头，将数字从 $123$ 改变为 $312$。

黑板上最初的数字是 $1$。请问最少需要多少次操作才能将黑板上的数字变为 $N$？如果无法将数字变为 $N$，则输出 $-1$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a positive integer $a$. Additionally, there is a blackboard with a number written in base $10$.
Let $x$ be the number on the blackboard. Takahashi can do the operations below to change this number.

• Erase $x$ and write $x$ multiplied by $a$, in base $10$.
• See $x$ as a string and move the rightmost digit to the beginning.
  This operation can only be done when $x \geq 10$ and $x$ is not divisible by $10$.

For example, when $a = 2, x = 123$, Takahashi can do one of the following.

• Erase $x$ and write $x \times a = 123 \times 2 = 246$.
• See $x$ as a string and move the rightmost digit 3 of 123 to the beginning, changing the number from $123$ to $312$.

The number on the blackboard is initially $1$. What is the minimum number of operations needed to change the number on the blackboard to $N$? If there is no way to change the number to $N$, print $-1$.

Constraints

• $2 \leq a \lt 10^6$
• $2 \leq N \lt 10^6$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$a$ $N$

Output

Print the answer.

Sample Input 1

3 72

Sample Output 1

4

We can change the number on the blackboard from $1$ to $72$ in four operations, as follows.

• Do the operation of the first type: $1 \to 3$.
• Do the operation of the first type: $3 \to 9$.
• Do the operation of the first type: $9 \to 27$.
• Do the operation of the second type: $27 \to 72$.

It is impossible to reach $72$ in three or fewer operations, so the answer is $4$.

Sample Input 2

2 5

Sample Output 2

-1

It is impossible to change the number on the blackboard to $5$.

Sample Input 3

2 611

Sample Output 3

12

There is a way to change the number on the blackboard to $611$ in $12$ operations: $1 \to 2 \to 4 \to 8 \to 16 \to 32 \to 64 \to 46 \to 92 \to 29 \to 58 \to 116 \to 611$, which is the minimum possible.

Sample Input 4

2 767090

Sample Output 4

111

update @ 2024/3/10 10:11:51