TITLE: What are the necessary conditions for UPC primes?
QUESTION [5 upvotes]: 0 68000 00027 7 is a UPC that the Hershey Company could use for some candy bar or other product. It happens that $6800000027$ is a prime number. But $68000000277$ is not. Likewise $6800000047$ is a prime number but quite obviously $68000000475$ is not. I'm sure that there exist GTIN-12 number that are prime which are also prime without the check digit, and maybe some of those numbers correspond to actual products you can buy at the supermarket or other store.
But what I want to know is this: what are the necessary (but not necessarily sufficient) conditions for such prime numbers?

REPLY [2 votes]: Don't give me the bounty, I just want to make some remarks that might be too long for comments.
The condition Barry gives in his answer was very useful to me in my search for such a pair that could theoretically correspond to a Hershey product, leading me to find $0$ $68000$ $00447$ $3$.
But before I use an online tool to check whether this corresponds to an actual product, I try the tool with a product I actually know to exist, one which I enjoyed very recently: Hershey's Pot of Gold, $0$ $68000$ $00113$ $7$ (do note that although $68000001137$ is a prime number, $6800000113$ is not, being divisible by $3$, $59$ and another prime). This one checks out: http://www.upcdatabase.com/item/068000113137
But no luck on $0$ $68000$ $00447$ $3$: http://www.upcdatabase.com/itemnotfound.asp?upc=0068000004473
Unfortunately the close neighbors search function is not available to me. Another website seems to suggest that Hershey has only one product with the $68000$ barcode, but I could have misunderstood. Maybe big companies can hold on to such large UPC blocks, I don't know.