2019/round3/integers_as_service.md

Looking for a new direction for his company, Carlos wants to break into the
most exciting tech field: digital rights management (DRM). In the world of
DRM, mega-corporations wage perpetual war against a vast sea of e-pirates,
modern-day Robin Hoods of ambiguous moral character.

One of the biggest players in this struggle is Sony, the well-known
copyrighter of numbers. Considering the speed with which Sony puts out new
products, it's no wonder they're having trouble coming up with enough secret
encryption keys to protect all of their intellectual property. Enter Carlos.

_ "Gone are the days of paying over-priced number theory PhDs to craft primes
by hand. I can make you a system that will generate all the numbers you need,
to your exact specifications." _

Sony wants to see a demonstration of Carlos's system before forking over
millions of dollars in consultancy fees. They have some new products in
development, each of which requires a secret key, **X**. For each key, Sony
has a list of **N** requirements. The _i_th requirement has an operator
character **Oi**, an integer value **Vi**, and an integer result **Ri**.

When **Oi** is 'G', the _i_th requirement states that the greatest common
divisor of **X** and **Vi** must be **Ri**. That is, `GCD(X, Vi) = Ri`.

When **Oi** is 'L', the _i_th requirement states that the least common
multiple of **X** and **Vi** must be **Ri**. That is, `LCM(X, Vi) = Ri`.

There is also a global requirement that all of Sony's secret keys must be
positive integers no larger than 1,000,000,000. Help Carlos build any positive
integer **X** consistent with all of these requirements, or determine that no
such integer exists.

### Input

Input begins with an integer **T**, the number of secret keys that Sony wants
Carlos to generate. For each number, there is first a line containing the
integer **N**. Then, **N** lines follow, the _i_th of which contains the
character **Oi**, and the integers **Vi** and **Ri**, all separated by spaces.

### Output

For the _i_th secret key, print a line containing "Case #_i_: " followed by a
single integer: your chosen value of **X**, or -1 if no valid integer **X**
exists.

### Constraints

1 ≤ **T** ≤ 250  
1 ≤ **N** ≤ 2,000  
**Oi** ∈ {'G', 'L'}   
1 ≤ **Vi**, **Ri** ≤ 1,000,000,000  

### Explanation of Sample

In the first case, `GCD(6, 4) = 2`, meaning that **X** = 6 satisfies the one
and only requirement. Note that, for this case and potentially other cases
below, **various other outputs will also be accepted**.

In the second case, there exists no valid integer **X** such that 1 ≤ **X** ≤
1,000,000,000 and `LCM(X, 4) = 2`.

In the third case, `GCD(24, 18) = 6`, `LCM(24, 40) = 120`, and `GCD(24, 20) =
4`, meaning that **X** = 24 is a valid choice.