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Second, using the natural number, 1-10, as either the numerator or denominator of a fraction, there is 100 fractions, 1/1, 2/1,...,10/1, 1/2, 2/2,...10/10. How many of these reduce to integers. I say 27, all of the ones (10), evens over 2 (5), multiples of 3 over three (3), multiples of 4 over 4 (2), 5 and 10 over 5 (2), and then 6, 7, 8, 9, 10 over themselves (5)

If we reduce all of these fractions, how many different numbers do we get? I say 10. There was ten to begin with and they all appear over 1.