CF1007B Pave the Parallelepiped

You are given a rectangular parallelepiped with sides of positive integer lengths $ A $ , $ B $ and $ C $ . Find the number of different groups of three integers ( $ a $ , $ b $ , $ c $ ) such that $ 1\leq a\leq b\leq c $ and parallelepiped $ A\times B\times C $ can be paved with parallelepipeds $ a\times b\times c $ . Note, that all small parallelepipeds have to be rotated in the same direction. For example, parallelepiped $ 1\times 5\times 6 $ can be divided into parallelepipeds $ 1\times 3\times 5 $ , but can not be divided into parallelepipeds $ 1\times 2\times 3 $ .

The first line contains a single integer $ t $ ( $ 1 \leq t \leq 10^5 $ ) — the number of test cases. Each of the next $ t $ lines contains three integers $ A $ , $ B $ and $ C $ ( $ 1 \leq A, B, C \leq 10^5 $ ) — the sizes of the parallelepiped.

For each test case, print the number of different groups of three points that satisfy all given conditions.

In the first test case, rectangular parallelepiped $ (1, 1, 1) $ can be only divided into rectangular parallelepiped with sizes $ (1, 1, 1) $ . In the second test case, rectangular parallelepiped $ (1, 6, 1) $ can be divided into rectangular parallelepipeds with sizes $ (1, 1, 1) $ , $ (1, 1, 2) $ , $ (1, 1, 3) $ and $ (1, 1, 6) $ . In the third test case, rectangular parallelepiped $ (2, 2, 2) $ can be divided into rectangular parallelepipeds with sizes $ (1, 1, 1) $ , $ (1, 1, 2) $ , $ (1, 2, 2) $ and $ (2, 2, 2) $ .