1

How would I be able to write the proposition -2n+1 is divisible by 5 if and only if 3n-4 is divisible by 5?

Stranger12345
  • 11
  • 1
    Write or prove? Note that $-2n+1=(3n-4)-5(n-1)$. – André Nicolas Dec 10 '13 at 18:44
  • Can you show me how to prove also? – Stranger12345 Dec 10 '13 at 18:51
  • I gave a hint above. (i) verify that what I wrote is correct. (ii) Suppose $5$ divides $3n-4$, say $3n-3=5q$. We show that $5$ divides $-2n+1$. By the above, $-2n+1=5q-5(n-1)=5(q-(n-1))$, and therefore $-2n+1$ is divisible by $5$. (iii) Suppose $5$ divides $-2n+1$. Show that $3n-4$ is divisible by $5$. Similar argument to the one in (ii), but it's your turn. – André Nicolas Dec 10 '13 at 18:58

1 Answers1

1

Probably something like this?

$$5 \mid -2n + 1 \iff 5 \mid 3n - 4.$$

John
  • 26,319