2012/round3/divisor_function_optimization.md

Let **d(N)** be the number of positive divisors of positive integer **N**.
Consider the infinite sequence **x(n) = d(n)a / nb, n = 1, 2, 3, …** where
**a** and **b** are fixed positive integers. It can be shown that this
sequence tends to zero. Hence it attains its maximum. Denote it by **p/q**
where **p** and **q** are co-prime positive integers. Your task is for given
**a** and **b** find **p** and **q** modulo **M = 109+7**. But to keep input
and output small you will be given tuples **(b1; b2; a1; a2; c)** and need to
calculate the sum of **(p mod M)** for all pairs **(a; b)** such that **b1 ≤ b
≤ b2**, **a1 ≤ a ≤ a2** and ** a ≤ c*b**, and the same sum for **q**-values.

### Input

The first line contains a positive integer **T**, the number of test cases.
**T** test cases follow. The only line of each test case contains five space
separated positive integers **b1, b2, a1, a2** and **c**.

### Output

For each of the test cases numbered in order from **1** to **T**, output "Case
#i: " followed by a space separated pair of integers: the sum of **(p mod M)**
for all pairs **(a; b)** mentioned above and the sum of **(q mod M)** for all
such pairs. Note that you need to find the sum of residues not the residue of
sum (see testcase 3 as a reference).

### Constraints

1 ≤ **T** ≤ 20  
1 ≤ **b1** ≤ **b2** ≤ 10,000  
1 ≤ **a1** ≤ **a2** ≤ 250,000  
1 ≤ **c** ≤ 25  
in each testcase the total number of pairs **(a; b)** for which the answer
should be calculated does not exceed 100,000