There are lots and lots of numbers three-digit numbers that are common multiples of 4 and 6. The first that comes to mind is 600. The lowest such number is 108. The highest is 996.
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Numbers divisible by 6 will have at least one 2 and one 3 as prime factors because 2 x 3 = 6. The same is true for 2, 4 and 8. Numbers divisible by 4 are also divisible by 2. Numbers divisible by 8 are also divisible by 4 and 2.
None of the numbers you listed are evenly divisible by 1000.
Numbers for which the sum of the digits is divisible by 9. This is also true for 3. There are other divisibility/multiple tests for other numbers (e.g., numbers that are divisible by 5 end in 5 or 0; numbers whose last two digits are divisible by 4 are divisible by 4)
All the numbers ending with 0,2,4,6,8 are divisible by 2.
Any number that ends in an even number (0, 2, 4, 6, 8) is divisible by 2.