I Hate Non-integer Number

AtCoder

IDabc262_d

Time2500ms

Memory256MB

Difficulty—

You are given a sequence of positive integers $A=(a_1,\ldots,a_N)$ of length $N$. There are $(2^N-1)$ ways to choose one or more terms of $A$. How many of them have an integer-valued average? Find the count modulo $998244353$. ## Constraints * $1 \leq N \leq 100$ * $1 \leq a_i \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $\ldots$ $a_N$ [samples]

Samples

Input #1

3
2 6 2

Output #1

6

For each way to choose terms of $A$, the average is obtained as follows:

*   If just $a_1$ is chosen, the average is $\frac{a_1}{1}=\frac{2}{1} = 2$, which is an integer.
    
*   If just $a_2$ is chosen, the average is $\frac{a_2}{1}=\frac{6}{1} = 6$, which is an integer.
    
*   If just $a_3$ is chosen, the average is $\frac{a_3}{1}=\frac{2}{1} = 2$, which is an integer.
    
*   If $a_1$ and $a_2$ are chosen, the average is $\frac{a_1+a_2}{2}=\frac{2+6}{2} = 4$, which is an integer.
    
*   If $a_1$ and $a_3$ are chosen, the average is $\frac{a_1+a_3}{2}=\frac{2+2}{2} = 2$, which is an integer.
    
*   If $a_2$ and $a_3$ are chosen, the average is $\frac{a_2+a_3}{2}=\frac{6+2}{2} = 4$, which is an integer.
    
*   If $a_1$, $a_2$, and $a_3$ are chosen, the average is $\frac{a_1+a_2+a_3}{3}=\frac{2+6+2}{3} = \frac{10}{3}$, which is not an integer.
    

Therefore, $6$ ways satisfy the condition.

Input #2

5
5 5 5 5 5

Output #2

31

Regardless of the choice of one or more terms of $A$, the average equals $5$.

API Response (JSON)

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    "name": "I Hate Non-integer Number",
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      "content": "You are given a sequence of positive integers $A=(a_1,\\ldots,a_N)$ of length $N$.   There are $(2^N-1)$ ways to choose one or more terms of $A$. How many of them have an integer-valued average? Find t",
      "description_type": "Markdown"
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      "memory_limit": 262144
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    "is_remote": true,
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    "sync_url": null,
    "sign": "abc262_d"
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      "statement_type": "Markdown",
      "content": "You are given a sequence of positive integers $A=(a_1,\\ldots,a_N)$ of length $N$.  \nThere are $(2^N-1)$ ways to choose one or more terms of $A$. How many of them have an integer-valued average? Find t...",
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