There is no such number. If X were a common multiple of 4 and 3 then a number of the form k*X (where k is any integer) would also be a multiple. There is no limit to how large k can be and so no greatest multiple.
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There is no "greatest common multiple" for any pair of numbers. If you meant greatest common factor, the answer is 3.
The greatest 4-digit number that is divisible by 16 is 9984
3
I assume you mean LCM, least common multiple. The least common multiple of 3 and 4 is 12. 12 is a multiple of 3. 12 is a multiple of 4. No smaller number is a multiple of both 3 and 4.
Common multiples of 2, 3 and 4 include 12, 24, 36 and 48. The greatest common multiple cannot be expressed as an integer.