Q. The Only Level TOO

Codeforces

IDCF10253Q

Time2000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

You've proven your worth and capabilities as a spy when you passed The Only Level. Your name isn't Mario but you fall down from a pipe. You find an elephant in a room. It speaks. You are not sure if you are expecting an elephant in this scenario, but the elephant gives you no time to be confused as it tells you: " The *digital sum* of an integer in base $b$ is the sum of its digits when it is written in base $b$. The *digital root* of an integer in base $b$ is the unique digit obtained by repeatedly computing the digital sum in base $b$ until only one digit remains. Let $f_b (k)$ be the digital root of the integer $k$ in base $b$. We say that the pair $(k, b)$ is *cool* if and only if $(f_b (0), f_b (k), f_b (2 k), \\dots, f_b ((b -1) k))$ is a permutation of $(0, 1, 2, \\dots, b -1)$. Let $K$ and $B$ be two sets of integers. Find the number of distinct cool pairs $(k, b)$ such that $k in K$ and $b in B$. " The first line of input contains two space-separated integers $| K |$ and $| B |$ denoting the sizes of the sets $K$ and $B$, respectively. The second line of input contains $| K |$ space-separated integers denoting the elements of the set $K$. The third line of input contains $| B |$ space-separated integers denoting the elements of the set $B$. *Constraints* $1 <= | K |, | B | <= 3 dot.op 10^5$ Every element of $K$ is a positive integer $<= 10^6$. Every element of $B$ is a positive integer $<= 10^6$. The elements of $K$ are distinct. The elements of $B$ are distinct. Output a single line containing a single integer denoting the answer. ## Input The first line of input contains two space-separated integers $| K |$ and $| B |$ denoting the sizes of the sets $K$ and $B$, respectively. The second line of input contains $| K |$ space-separated integers denoting the elements of the set $K$. The third line of input contains $| B |$ space-separated integers denoting the elements of the set $B$. *Constraints*$1 <= | K |, | B | <= 3 dot.op 10^5$Every element of $K$ is a positive integer $<= 10^6$.Every element of $B$ is a positive integer $<= 10^6$.The elements of $K$ are distinct.The elements of $B$ are distinct. ## Output Output a single line containing a single integer denoting the answer. [samples]

**Definitions** Let $ f_b(k) $ denote the digital root of $ k $ in base $ b $, defined as the repeated sum of digits in base $ b $ until a single digit is obtained. **Given** - Finite sets $ K \subset \mathbb{Z}^+ $, $ B \subset \mathbb{Z}^+ $, with $ |K|, |B| \leq 3 \cdot 10^5 $. - For each $ k \in K $, $ k \leq 10^6 $. - For each $ b \in B $, $ b \leq 10^6 $. **Constraint** For a pair $ (k, b) $ to be *cool*, the sequence $$ (f_b(0), f_b(k), f_b(2k), \dots, f_b((b-1)k)) $$ must be a permutation of $ \{0, 1, 2, \dots, b-1\} $. **Objective** Compute the number of distinct cool pairs $ (k, b) \in K \times B $.

API Response (JSON)

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    "name": "Q. The Only Level TOO",
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      "content": "You've proven your worth and capabilities as a spy when you passed The Only Level. Your name isn't Mario but you fall down from a pipe. You find an elephant in a room. It speaks. You are not sure ",
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      "time_limit": 2000,
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