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Okay, well I'm just curious what the lowest possible IV a Pokemon can have, also is it possible to have a perfect 31 IV Pokemon legitimately?

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The lowest possible IV in a stat is zero. I know because I have bred Pokémon with 0IVs in a stat.

As for your second question, it is totally possible to obtain 6IV Pokémon legitimately. While it is very rare, it can be done several ways.

1. Lady Luck
You could always get lucky and randomly encounter a 6IV Pokémon. However, this is ver, very, very unlikely. I believe I read somewhere that the chances are 8 times less than finding a Shiny (but don't quote me on that — the point is that it is extremely rare).

2. Special Snowflakes
There are some Pokémon that are guaranteed to have a minimum of three perfect IVs. These are Legendary Pokémon like Kyogre and Mewtwo, Mythical Pokémon like Genesect and Magearna, and Ultra Beasts like Buzzwole and Xurkitree. While the other 3IVs are still up to chance, it is much more likely to catch a 6IV "special" Pokémon than a random wild Pokémon. Keep in mind that it is still extremely rare.

3. Save Our Souls
You can also encounter Pokémon with perfect IVs by SOS chaining in Sun and Moon (also the DexNav in ORAS, but I will be talking about the newest games.) The higher the chain of Pokémon that call for help, the likelier they are to have perfect IVs. The maximum is 4IVs after hitting a chain of 30. For more, read here. Keep in mind that the other two IVs are random, but there is a small chance of all six being perfect.

4. Birds and Bees and Eggs
The last three tips have been either "hope for perfect IVs" or "get as many guaranteed perfect IVs as possible, and hope for the rest." However, there is one way to get six IVs that is much more likely to succeed, and that is breeding. If you breed two Pokémon with 5-6 perfect IVs while holding a Destiny Knot, the chance of a 6IV offspring is much higher than the other methods. Getting two 5IV parents isn't that difficult, it just requires patience and persistence. If you manage to get your hands on a 6IV Ditto, it's even better. You see, the Destiny Knot passes down 5 random IVs from either parent. If you breed two parents with lots of perfect IVs, the offspring will also have a lot of perfect IVs. By breeding 5IV + 5IV, 5IV + 6IV, or 6IV + 6IV, you maximize your chances of hatching a 6IV offspring. Be warned, however, that it is still difficult. I have bred countless 5IV Pokémon from a 5IV parent and a 6IV Ditto and have only hatched two 6IV offspring. Patience is key. Also keep in mind that most Pokémon only need 5IVs.

I hope this answers you question!

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>In Generation III, the IV system was completely overhauled; now, IVs range from 0-31 rather than 0-15, with HP and Special Defense getting their own independent IVs.

The lowest possible amount of IVs that a Pokemon can legitimately have in a stat is zero. To answer your second question, yes, it is possible for a Pokemon to legitimately have 31 IVs in all stats.

Sources: This and knowledge.

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