D - Multiply and Rotate

Score : $400$ points

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.

3 72

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 $4$.

2 5

-1

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

2 611

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.

2 767090

111