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Galileo

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I thought the easiest way to do it was to show that [itex]1 \in I^n+J^m[/itex] given that there exist [itex]i\in I[/itex] and [itex]j\in J[/itex] such that [itex]i+j=1[/itex]. But I haven't had much luck yet. Any hint would be appreciated.