Last digit of 1/3

Question

I stumbled upon a very bizarre result when playing with wolfram alpha. According to it, the last digit of $\frac{1}{3} = 7$. Is that true? If yes, is there a mathematical argument for such a result?

The reason I require an explanation for such a statement is that the impression is that the theoretical definition of infinite decimal expansion suggests that the last digit should be 3. Moreover, if we force a different number, (mostly used by calculators) to terminate the decimal expansion in the manner $0.333333....4$, since it's more "sensible" for a calculator, even such an argument motivates the use of $4$ as the last digit. So why $7$?

Answer 1

If the last digit of $n$ is $d$, then this means $n\equiv d \pmod{10}$.

Now $3\cdot 7 = 21 \equiv 1 \pmod{10}$, therefore it makes sense to say $1/3\equiv 7 \pmod{10}$ (note that $\gcd(3,10)=1$, therefore it actually makes sense to speak of an inverse of $3$ modulo $10$).

Interpreting “last digit of” as “the single digit number which the number is equivalent to module $10$” then gives the Wolfram result.

Answer 2

It's likely Wolfram Alpha computed the modular inverse of $3 \bmod 10$, which is $7$.