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A jamcoin is a string of N ≥ 2 digits with the following properties:
1and the last digit is
Not every string of
1s is a jamcoin. For example,
101 is not a jamcoin; its interpretation in base 2 is 5, which is prime. But the string
1001 is a jamcoin: in bases 2 through 10, its interpretation is 9, 28, 65, 126, 217, 344, 513, 730, and 1001, respectively, and none of those is prime.
We hear that there may be communities that use jamcoins as a form of currency. When sending someone a jamcoin, it is polite to prove that the jamcoin is legitimate by including a nontrivial divisor of that jamcoin's interpretation in each base from 2 to 10. (A nontrivial divisor for a positive integer K is some positive integer other than 1 or K that evenly divides K.) For convenience, these divisors must be expressed in base 10.
For example, for the jamcoin
1001 mentioned above, a possible set of nontrivial divisors for the base 2 through 10 interpretations of the jamcoin would be: 3, 7, 5, 6, 31, 8, 27, 5, and 77, respectively.
Can you produce J different jamcoins of length N, along with proof that they are legitimate?
The first line of the input gives the number of test cases, T. T test cases follow; each consists of one line with two integers N and J.
Note that, unusually for a Code Jam problem, you already know the exact contents of each input file. For example, the Small dataset's input file will always be exactly these two lines:
So, you can consider doing some computation before actually downloading an input file and starting the clock.
For each test case, output J+1 lines. The first line must consist of only
Case #x:, where
x is the test case number (starting from 1). Each of the last J lines must consist of a jamcoin of length N followed by nine integers. The i-th of those nine integers (counting starting from 1) must be a nontrivial divisor of the jamcoin when the jamcoin is interpreted in base i+1.
All of these jamcoins must be different. You cannot submit the same jamcoin in two different lines, even if you use a different set of divisors each time.
1 6 3
Case #1: 100011 5 13 147 31 43 1121 73 77 629 111111 21 26 105 1302 217 1032 513 13286 10101 111001 3 88 5 1938 7 208 3 20 11
In this sample case, we have used very small values of N and J for ease of explanation. Note that this sample case would not appear in either the Small or Large datasets.
This is only one of multiple valid solutions. Other sets of jamcoins could have been used, and there are many other possible sets of nontrivial base 10 divisors. Some notes:
110111could not have been included in the output because, for example, it is 337 if interpreted in base 3 (1*243 + 1*81 + 0*27 + 1*9 + 1*3 + 1*1), and 337 is prime.
010101could not have been included in the output even though
10101is a jamcoin, because jamcoins begin with
101010could not have been included in the output, because jamcoins end with
110011is another jamcoin that could have also been used in the output, but could not have been added to the end of this output, since the output must contain exactly J examples.
100011could not have been either 1 or 35, because those are trivial divisors of 35 (
100011in base 2).