Until you get good at "seeing" GCF's it helps to look at each term in fully factored form:
15x2y3   = 3 * 5         * x * x * y * y * y
15xy3    = 3 * 5         * x     * y * y * y
18x2y2z3 = 3     * 3 * 2 * x * x * y * y     * z * z * z
I have used spacing to make it is easier to see all the factors which are in all three terms. We can see that there is one 3, one x and two y's in all three terms. So the GCF = 3*x*y*y =