Naturally? No. By HM? Yes.
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Yes. HM01 - Cut can be used to tutor a Sandshrew.
In Generation I (Red, Blue, Green, Yellow), the Pokemon who can use cut are: Bulbasaur, Ivysaur, Venusaur, Charmander, Charmeleon, Charizard, Beedrill, Sandshrew, Sandslash, Bellsprout, Weepinbell, Victreebell, Tentacool, Tentacruel, Farfetch'd, Krabby, Kingler, Lickitung, Tangele, Scyther, Pinsir, and Mew. In Pokemon Yellow, Kabutops, Diglett, and Dugtrio can also learn Cut.
only Pokemon with sharp claws can learn the hm cut
No, it is not able to learn cut.
yes luxio can learn cut luxray can learn cut and shinks can almost every Pokemon can earn cut