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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Find the integer numbers m for which the fraction:

is an integer. How many such numbers have you found?

The number 4m+3 must be a divisor of 64. Therefore by the theorem of integer division we have:

where k is any integer. However, 64 has only factors of 2, as can be seen from its factoring. At the same time, 4m+3 can only take odd values, since 4m is necessarily even and we add an odd number to it. If 4m+3 is to be a factor of 64 it can only be equal to 1:

There is one solution.