R. The Only Level 3

Codeforces

IDCF10253R

Time6000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

You've passed the Only Level TOO. Your name isn't Mario but you fall down from a pipe. You find an elephant in a room. It speaks. You now know where this is going. The elephant speaks: " 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 $(0, 1, 2, \\dots, b -1)$. Given two integers $k '$ and $b '$, find the number of cool pairs $(k, b)$ such that $1 <= k <= k '$ and $2 <= b <= b '$, modulo $10^6$. " The input consists of a single line containing two space-separated integers $k '$ and $b '$. *Constraints* Output a single line containing a single integer denoting the answer. ## Input The input consists of a single line containing two space-separated integers $k '$ and $b '$. *Constraints* $1 <= k ' <= 5 dot.op 10^9$ $2 <= b ' <= 5 dot.op 10^9$ ## 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. **Constraints** Given integers $ k' \in \mathbb{Z}^+ $, $ b' \in \mathbb{Z}^+ $ with $ 1 \leq k' \leq 10^6 $, $ 2 \leq b' \leq 10^6 $. **Objective** Count the number of pairs $ (k, b) $ such that: - $ 1 \leq k \leq k' $, - $ 2 \leq b \leq b' $, - The sequence $ \left( f_b(0), f_b(k), f_b(2k), \dots, f_b((b-1)k) \right) $ is a permutation of $ \{0, 1, 2, \dots, b-1\} $. Output the count modulo $ 10^6 $.

API Response (JSON)

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