Let a and b be the numbers, both numbers must be even if 2 is the GCF, so let a=2m and b=2n where m and n are integers. m and n have no other factors in common, so mn is their LCM and the LCM of a and b is 2mn=60 and mn=30, which has factors 1 and 30, 2 and 15, 3 and 10, 5 and 6.
So a and b can be 2 and 60, 4 and 30, 6 and 20, 10 and 12.
Leillah is wrong in thinking that a and b can only be 10 and 12.