Nim

AtCoder

IDabc317_f

Time2000ms

Memory256MB

Difficulty—

You are given integers $N,A_1,A_2,A_3$. Find the number, modulo $998244353$, of triples of positive integers $(X_1,X_2,X_3)$ that satisfy all of the following three conditions. * $1\leq X_i \leq N$ for every $i$. * $X_i$ is a multiple of $A_i$ for every $i$. * $(X_1 \oplus X_2) \oplus X_3 = 0$, where $\oplus$ denotes bitwise xor. What is bitwise xor? The bitwise xor of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows. * When $A \oplus B$ is written in binary, the $2^k$s place ($k \geq 0$) is $1$ if exactly one of the $2^k$s places of $A$ and $B$ is $1$, and $0$ otherwise. For instance, $3 \oplus 5 = 6$ (in binary: $011 \oplus 101 = 110$). ## Constraints * $1 \leq N \leq 10^{18}$ * $1 \leq A_i \leq 10$ * All input values are integers. ## Input The Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $A_3$ [samples]

Samples

Input #1

13 2 3 5

Output #1

4

Four triples $(X_1,X_2,X_3)$ satisfy the conditions: $(6,3,5),(6,12,10),(12,6,10),(12,9,5)$.

Input #2

1000000000000000000 1 1 1

Output #2

426724011

Input #3

31415926535897932 3 8 4

Output #3

759934997

API Response (JSON)

{
  "problem": {
    "name": "Nim",
    "description": {
      "content": "You are given integers $N,A_1,A_2,A_3$. Find the number, modulo $998244353$, of triples of positive integers $(X_1,X_2,X_3)$ that satisfy all of the following three conditions. *   $1\\leq X_i \\leq N$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc317_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given integers $N,A_1,A_2,A_3$. Find the number, modulo $998244353$, of triples of positive integers $(X_1,X_2,X_3)$ that satisfy all of the following three conditions.\n\n*   $1\\leq X_i \\leq N$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments