Let N(x) be the number 122....221 where the digit 2 occurs x times.
in the past
we have determined the highest power of 11 that divides N(2001) is 11^3.
What is the smallest x for N(x) to be a multiple of 11^3? What about multiples of 11^4 and 11^5?
Mike wanted to buy a bike, and he found one he liked at $20, so he bought it. The next day, a friend saw it and offered $30 dollars for it so Mike agreed. He later bought it back for $40. Again someone saw it and offered $50 and again Mike sold it. Mike finally rebought the bike for good for $60. Mike then took it to a bike show where a man told him the bike was worth $40.
Deciding to keep it, did Mike gain or lose money on the transactions? Count the bikes worth in the problem.