Let "d" be the number of diamonds, "r" the number of rubies, and "e" the number of emeralds.
Saying "the ratio of diamonds to rubies is 5/6" means that d/r= 5/6 or d= (5/6)r.
Saying "the ratio of rubies to emeralds is 8/3" means that r/e= 8/3 or r= (8/3)e.
In order that d be a whole number, r must be a multiple of 6. In order that r be a whole number, e must be a multiple of 3. We can pick up a factor of 2 by multiplying by 8 but to get the remaining factor of 3 in 6= 2(3), e/3 must be a multiple of 3 which means that e itself must be a multiple of 3(3)= 9.
The smallest possible value of e, then, is 9. Now you can find both r and d and then the total number of jewels.