23008번 - Alien Generator 서브태스크다국어

Astronauts have landed on a new planet, Kickstartos. They have discovered a machine on the planet: a generator that creates gold bars. The generator works as follows. On the first day, an astronaut inputs a positive integer K into the generator. The generator will produce K gold bars that day. The next day, it will produce K+1, the following day, K+2, and so on. Formally, on day i, the generator will produce K+i−1 gold bars.

However, the astronauts also know that there is a limitation to the generator:

if on any day, the generator would end up producing more than G gold bars in total across all the days, then it will break down on that day and will produce 0 gold bars on that day and thereafter. The astronauts would like to avoid this, so they want to produce exactly G gold bars.

Consider K=2 and G=8. On day 1, the generator would produce 2 gold bars. On day 2, the generator would produce 3 more gold bars making the total gold bars is equal to 5. On day 3, the generator would produce 4 more gold bars which would lead to a total of 9 gold bars. Thus, the generator would break on day 3 before producing 4 gold bars. Hence, the total number of gold bars generated is 5 in this case.

Formally, for a given G, astronauts would like to know how many possible values of K on day 1 would eventually produce exactly G gold bars.

The first line of the input gives the number of test cases, T. T lines follow.

Each line contains a single integer G, representing the maximum number of gold bars the generator can generate.

For each test case, output one line containing Case #x: y, where x is the test case number (starting from 1) and y is the number of possible values of K on day 1 that would eventually produce exactly G gold bars.

• 1 ≤ T ≤ 100

• 1 ≤ G ≤ 10⁴.

• 1 ≤ G ≤ 10¹² for at most 20 test cases.
• For the remaining cases, 1 ≤ G ≤ 10⁴.

For Sample Case #1, there are 2 possible values of K (1, 10) that would eventually produce exactly 10 gold bars. For K=1, we will have 1+2+3+4=10 gold bars after 4 days, and for K=10, we will have 10 gold bars after just 1 day.

For Sample Case #2, there are 4 possible values of K (8, 23, 62, 125) that would eventually produce exactly 125 gold bars.