Well, there's no right or wrong when it comes to Pokeballs. I suggest Great Balls, Ultra Balls or Dusk Balls since Axew is one of the more harder Pokemon to catch.
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by throwing it
the ultra ball is what i used to catch it...but you could use the master ball after you get it
master. no other ball can hold it.
Yes, the Ultra Ball has a 4x catch rate than a Premier Ball, which has the same catch rate as a normal Poke Ball. It is commonly used to catch rare or Legendary Pokemon.
If you are in battle with another trainer you cannot catch that Pokemon that belongs to them. But if you find out where that Pokemon's habitat is (by using the pokedex) then you can go and catch it with a poke ball, great ball, ultra ball etc...