Production Conundrum
There are two machines:
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machine 1 which produces k+1 shoes every minute. (That is, one shoe comes out of the machine every 1/(k+1) th of a minute.)
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machine 2 which produces k shoes every minute. (That is, one shoe comes out of the machine every 1/k th of a minute.) where k is a positive integer.
Both machines produce alternatingly right and left shoes (starting with the left shoe). They are switched on at the exact same time. When machine 1 has produced 2017 more shoes than machine 2, how often did it occur that a left shoe came out of each machine at the exact same moment?
The answer is 0.
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1 solution
Since gcd(k,k+1) = 1, the machines only produce a shoe at the same time at 1 minute, 2 minutes etc. However, one of k and k+1 is even such that at each minute (the only points in time when both machines produce a shoe at the same time) will have produced an even number of shoes. Because the first produced shoe is a left one, one of the machines will produce only right shoes at each minute. Hence, the answer is 0.