Does the EAN-13 bar code show the country of origin?

No, the GTIN (the bar code number) does not imply any significance.

A GTIN starts with the GS1 Prefix of the GS1 Member Organisation that allocated a GS1 Company Prefix to a company to allow them to generate GTINs and bar code their products. This does not mean that the product was manufactured in a specific country or by a specific manufacturer, it may have been produced anywhere in the world.

You can type the full number into the GEPIR system and find out the company information. For more information please try 

http://gepir.gs1.org/v31/xx/

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