Fermat's little theorem.

An example says these two lines
x^3+2^3=1(mod3) is the problem posed
OK so far
Then points out that 2^3= 2 (mod 3) FLT, OK here too

then we get "therefore x^3=2(mod3)". I do not get that jump.
Could someone fill in the gap for me?

Reply 1

RDKGames

Study Forum Helper

Original post by nerak99

So by FLT, we have

x^3 + 2 \equiv 1 \pmod{3}

Subtract 2 from both sides:

x^3 \equiv -1 \pmod{3}

But since

-1 \equiv 2 \pmod{3}

the result follows.

(edited 5 years ago)

Reply 2

nerak99

BTW, the original Q is Screen Shot 2019-02-25 at 18.03.02.png

Reply 3

DFranklin

Original post by nerak99

BTW, the original Q is Screen Shot 2019-02-25 at 18.03.02.png

So, FWIW, I would go: by FLT, (x+2)^3 = (x+2) mod 3, so (x+2) = 1 mod 3. and then adding 1 to both sides gives the result.

It's also absolutely fine to simply go: x = 0 doesn't work, x = 1 doesn't work, x = 2 does work, so x = 2 mod 3, which is almost certainly the easiest option here...