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During a tournament, seven football teams, three European, three South American, and one from Africa, scored a total of 89 goals.

The number of goals scored by the African squad was relatively prime (i.e. had no common divisor) with that of any of the other six teams.

On the other hand, the numbers of the goals scored by any two European teams did have a common divisor greater than one, and so did those of any two South American teams.

However, except for two instances (involving altogether four teams), the numbers of goals scored by a European team and a South American one were always relatively prime.

How many goals did the African team score during the tournament?

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The answer is

17; the scores of the other teams (separated by continent) are 5, 10, 15 resp. 7, 14, 21.

How did I get to this?

I assumed the European scores had one common divisor, which did not divide the South American scores, and vice versa. If the divisor of one of the groups was even, the other group's scores all needed to be odd, making the African score even (otherwise the sum can't be 89). But then the African score isn't relatively prime with the even ones.
Next I tried 3 and 5, but the scores got too high (5, 10, 35 vs. 3, 6, 21), which would result in 9 for the African score.
5 and 7 worked better and resulted in the final answer. 10 and 14 share a common divisor, 15 and 21 as well but the rest don't. There are 17 goals left for the African team, which fortunately happens to be prime.

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