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Alice writes down all 7-digit numbers using the digits 1, 2, 3, 4, 5, 6, and 7 exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
You have N bags. Bag 1 has a black ball, Bag 2 has a black ball and a white ball, Bag 3 has a black ball and two white balls, and so on. Bag N has a black ball and N-1 white balls. You pick a ball from each bag at random and record the numbers of the bags that you picked a black ball from. For example, if you had 100 bags, then your sequence might be 1, 2, 3, 10, 14, 37. Call the last number in your sequence X. Prove that X is a random number from 1 to N with a uniform distribution.
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