Description

Given $n, m$, find how many pairs of positive integers $(i, j)$ satisfy the following system of equations:

$$\begin{cases} i+j=n \\ \lfloor\frac{i}{j}\rfloor+\lceil\frac{j}{i}\rceil=m \end{cases}$$

In the above, $\lfloor x\rfloor$ denotes rounding $x$ down, and $\lceil x\rceil$ denotes rounding $x$ up.

Input Format

This problem contains multiple test cases in a single input file.

The first line contains an integer $T$, indicating the number of test cases.

The next $T$ lines each contain two integers $n, m$, with the meaning as described in the statement.

Output Format

Your output should contain $T$ lines.

For each test case, output one integer per line representing your answer.

2
2 2
2 1

1
0

1
6 2

2

Hint

Sample Explanation

When $n=m=2$, only $(1,1)$ satisfies the conditions.
When $n=2,m=1$, there is no solution.
When $n=6,m=2$, only $(2,4)$ and $(3,3)$ satisfy the conditions.

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Constraints

This problem uses bundled testdata.

• Subtask 1 (10 pts): $T, n, m\le 500$.
• Subtask 2 (40 pts): $T, n, m\le 5000$.
• Subtask 3 (5 pts): $m=1$.
• Subtask 4 (5 pts): $m>n$.
• Subtask 5 (5 pts): $m\in[n-1,n]$.
• Subtask 6 (35 pts): no special constraints.

For $100\%$ of the testdata, $1\le T\le 10^5$, $1\le n,m\le 10^{7}$.

Translated by ChatGPT 5