Using Euclid's Algorithm prove..

Question

Using Euclid's Algorithm prove that the fraction $\frac{24n+5}{18n+4}$ is in lowest terms.

Is this solution going to be correct as a proof?

Thanks for help!

Answer 1

I believe your equations aren't written correctly.

Should be $24n + 5 = 1 \times (18n + 4) + (6n + 1)$, and that the third equation is not neccesary. Other than that, it's correct.

Answer 2

Yes, that is correct after fixing the first equation. Another way is to eliminate $\,n\,$ by using $\,24/18 = \color{#0a0}4/\color{#c00}3\ $ so $\ \color{#0a0}4(18n\!+\!4)- \color{#c00}3(24n\!+\!5)=1\ $ yields the Bezout identity for the gcd.