What is(are) the intermediate argument(s) behind the following lines?

\begin{align*} 3t&\equiv 4\mod8\\ \implies t&\equiv4\mod8 \end{align*}

Found it in a number theory text.

  • $\begingroup$ What is $3^{-1} \pmod 8$? $\endgroup$ – user296602 Oct 17 '17 at 20:37
  • $\begingroup$ 3t for t=4 it makes 12t and 12t mod 8 is 4 $\endgroup$ – Isham Oct 17 '17 at 20:39

$3t \equiv 4\mod 8$

$3t \equiv 4 + 8\mod 8$

$3t\equiv 12\mod 8$

multiplying by the multiplicative inverse of 3 :

$t \equiv 4\mod 8$

It's modulo 8 because $gcd(3,8)=1$ so we keep the modulo 8.

  • 1
    $\begingroup$ It might be worth mentioning why "dividing" is allowed here at all since it often isn't. It would also probably be better to refer to it as "multiplying by the multiplicative inverse of $3$" rather than "dividing by $3$" to avoid confusion. $\endgroup$ – JMoravitz Oct 17 '17 at 20:46
  • $\begingroup$ You right @JMoravitz I will correct it $\endgroup$ – Isham Oct 17 '17 at 20:48

Try multiplying both sides of the equation by $3$. What is the result modulo $8$?


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