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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.
Two positive integers x and x+3 can have:
For the numbers to have a common divisor of 2:
is not possible since, if x is even, x+3 must be odd, and conversely. They cannot both be even.
The same applies to divisibility by 4. Since only of the numbers can be even, they cannot both be multiples of 4.
For the numbers to have a common divisor of 3:
it is possible, since the numbers can be simultaneously multiples of 3.
Analogously, they cannot be both multiples of 5: