Triangles in a Grid

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Chef has a grid with $N$ equally spaced horizontal lines and $M$ equally spaced vertical lines. The points where a horizontal and a vertical line intersect are called grid points. Chef draws a triangle at random on this grid such that the vertices of the triangle lie on grid points and the area of the triangle is nonzero. All triangles satisfying this criteria are equally likely to be drawn by Chef. Calculate the [expected number](https://en.wikipedia.org/wiki/Expected_value#Definition) of grid points on the perimeter of this triangle, including the three vertices. It is guaranteed that it will be possible to express this value as an irreducible fraction $\frac{p}{q}$ where $q$ is coprime to $10^9 + 7$. Let $r$ be the [modular mutiplicative inverse](https://en.wikipedia.org/wiki/Modular_multiplicative_inverse) of $q$ with respect to modulus $10^9 + 7$. Output the answer as $p \cdot r \pmod{10^9 + 7}$. ###Input:  The first and only line contains 2 integers $N$ and $M$. ###Output:  Output in a single line the answer $p \cdot r \pmod{10^9 + 7}$. ###Constraints  $2 \leq N, M \leq 10^6$ ###Sample Input 1: 2 2 ###Sample Output 1: 3 ###Sample Input 2: 2 3 ###Sample Output2: 333333339 ###Explanation:  **Sample 1**: The are 4 eligible triangles with vertices on grid points and nonzero area. All of them have 3 grid points on their perimeter.  **Sample 2**: There are 12 triangles that have 3 grid points on the perimeter and 6 triangles that have 4 grid points on the perimeter. The expected value can be calculated as $3 \frac{12}{18} + 4 \frac{6}{18} = \frac{10}{3}$. So $p=10$, $q=3$ and $r=333333336$. The answer is $10 \cdot 333333336 \pmod{10^9 + 7} = 333333339$.Author:  meooow 
Editorial  https://discuss.codechef.com/problems/CLTNGL 
Tags  cole2018, hard, meooow, mobius 
Date Added:  5042018 
Time Limit:  2 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.5, PYPY, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, rust, SCALA, swift, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, kotlin, PERL6, TEXT, SCM chicken, CLOJ, COB, FS 
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