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Ex - Popcount Sum

Score : $600$ points

Problem Statement

Find the sum of popcounts of all integers between $1$ and $N$, inclusive, such that the remainder when divided by $M$ equals $R$.
Here, the popcount of a positive integer $X$ is the number of $1$s in the binary notation of $X$, that is, the number of non-negative integers $k$ such that the $2^k$s place is $1$.
For each input, process $T$ test cases.

Constraints

• $1 \leq T \leq 10^5$
• $1 \leq M \leq N \leq 10^9$
• $0 \leq R < M$
• All values in the input are integers.

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Input

The input is given from Standard Input in the following format. The first line is as follows:

$T$

$T$ test cases follow. Each of them is given in the following format:

``` $N$ $M$ $R$ ```

Output

Print $T$ lines. The $i$-th line should contain the answer to the $i$-th test case.

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2
12 5 1
6 1 0

6
9

In the $1$-st test case, the popcount of $1$ is $1$, the popcount of $6$ is $2$, and the popcount of $11$ is $3$, so $1+2+3=6$ should be printed.
In the $2$-nd test case, the popcount of $1$ is $1$, $2$ is $1$, $3$ is $2$, $4$ is $1$, $5$ is $2$, and $6$ is $2$, so $1+1+2+1+2+2=9$ should be printed.