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I just got my iMac through the education store. You can get both the gift card and the discount price. Also, no id is required. You just need to be going to a school that's in apple's list. I'm not sure how exactly they confirm it but that's all you need to do.
 
If you are looking to get a computer via edu through US apple, you need to be attending a US school regardless if you are a citizen or international student. The US apple store will not give discounts to foreign schools, though if you do it online, there's no stopping you from choosing a random school to get the discount but you need a US credit card.

If you try to do this in the store, you will likely be denied.
 
Is there not an apple store online for your country? I'm in Canada for example and the list of schools changed to Canadian ones for me.

Yeah, I am from Denmark and the list of schools also changed to Danish ones for me. Also you don't need anything online to get both the discount and gift card.
 
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