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Question

Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns. Phil buys a number of these packs for a barbecue. After the barbecue, Phil finds that he has 4 hot dogs left over. What is the SECOND smallest number of packs of hot dogs he could have bought?

Accepted Answer

Let n be the number of packs of hot dogs that Phil bought. Then 10n 4 8, which reduces to 2n 4 8. This congruence tells us that 2n = 8k + 4 for some integer k, or n = 4k + 2. The second smallest positive integer of this form is 6.