I have given as much information as I could about primoradic here.
In primoradic , multiples of 7 which are not multiples of 2, 3 or 5 end by (...:0:1:0:1), (...:1:3:0:1), (...:2:2:2:1), (...:3:0:0:1), (...:3:4:2:1), (...:4:2:0:1), (...:5:1:2:1) or (...:6:3:2:1). Let us apply that to an historical example. Consider a number studied by Blaise Pascal, N=287.542.178. Let $N1=N/2=143.771.089$. In primoradic, $$N1=(14:15:10:7:6:1:3:0:1)\equiv49 \pmod {210}$$ It was sufficient to have $(...:1:3:0:1)$ to know that Pascal's number is divisible by 7. By generalization, you have a divisibility rule for any prime number. For $p_{n+1}$, you have $Card((Z/p_n\#)^\times)$ endings. For example, for $11$, $(...:7:3:2:0:1)$ is one of the $48$ endings.
Notice : $$N1=(14:15:10:7:6:1:3:0:1)=1+3 \times6+1\times{30}+6\times{210}+7\times{2310}+10\times30030+15\times510510+14\times9699690$$ To ask a precise question : you could find a number like Blaise Pascal's one, which is a multiple of $7$ and write it in primoradic. Or a multiple of any prime number as 107 (see here)