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Problem 1988 MathHomeWork

## Problem Description

Brother LYG has learned a new function called function “F(n)”, where F(n) equals to the number of factors of n(including 1 and n). For example, F(1) = 1, F(4) = 3…Now if you ask LYG to calculate F(n), he may soon tell you the answer. If you are evil enough and ask him about the sum of the first K F(i)( F(1) + F(2) + .. + F(K) = G(k)), I believe out superbrother LYG will soon tell you the answer too! However, an evil math teacher gives LYG one non-negative integer M, and want our superbrother to tell us if there exist one integer K such that G(k) = M.

## Input

In the first line there is an integer T, indicates the number of test cases. (T <= 100) In each case, the first line contains only one integer M. (1 <= M <= 10^9).

## Output

For each case, output “Case idx: “ first where idx is the index of the test case start from 1, then if there exist such integer K, just output K, otherwise just output “impossible!” instead!

3 1 2 3

## Sample Output

Case 1: 1 Case 2: impossible! Case 3: 2

## Source

FOJ有奖月赛-2010年11月

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