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Professor Hornswoggle challenged his students to determine a mystery number. He gave them the following thirty-one clues:
1) The number is divisible by 2
2) The number is divisible by 3
3) The number is divisible by 4
$\vdots$
30) The number is divisible by 31
31) Exactly two of the clues above are incorrect and were given consecutively.
It turns out that it is not possible to determine the professor’s number based on the clues given above, but which two consecutive clues were incorrect?
Bonus: Determine the smallest positive number that satisfies the clues that are correct.
***December Solutions are due by December 14***