The best way to solve for common multiples is to use a Venn diagram. The question does not ask for the Lowest Common Multiple (LCM), yet seeks the first three common multiples. As such a Venn diagram will show this to be 114, 228, 342. As 38 is a multiple of 19, 19 can be disregarded. It will look like this:
6 12 18 24…
114, 120, 126…
6 210, 216, 222, 228…
330, 336, 342
38 76 114 152
38 190 228 266 304
342
Again, because 38 is a multiple of 19, disregard the 19 (it will always go into the multiples of 38). A Venn diagram usually has bars on either side of the multiples but you get the idea. Find all multiples of the numbers in question until you find common multiples. Finding the first three just requires additional multiples of each number until the first three have been found as in the example above.
The above example will always show you the multiples. Once you understand this you can solve these problems in an easier format:
6*19= 114 thus, because 19 is half of 38; 3*38= 114
6*38= 228 again 19 is half of 38, thus it will factor into 228 (228/19= 12)
Such that if 114 is the lowest common multiple of 6, 19, and 38; adding it to any subsequent multiple of 38 will net you common multiples of all three numbers ie (228+114= 342 your third common multiple). If you needed the fourth common multiple it would simply be 342+114= 456. A simple check shows this to be true; 456/6= 76 and because it factors evenly 456 is your fourth common multiple.
9/4/2013

Joshua F.