you can get HM 1 cut at cutter`s house in Rusboro city.
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get cut from the cut master in the city w/ the first gym and Devon inc
Zigzagoon, Poocheyna, Treeko, Chikorita, Linoone
firstly use cut then by using secret power (its a TM)
your Pokemon CANT learn cut in ruby, sapphire or emerald. you have to trade a Pokemon that's holding the TM cut (from sinnoh region game, platinum, diamond and pearl) then take it off them and they can learn cut.
In Rustboro City left from the Pokemon Center is a house. Go inside there should be a guy and talk to him.