If $mn+1$ is a multiple of $24$, prove that $m+n$ is also a multiple of 24.

Question

Please use UK pre-uni methods only (at least at first). Thank you.

Answer 1

Two facts:

• If $m$ is odd, then $8\mid m^2-1=(m-1)(m+1)$ because $m-1,m+1$ are consecutive even integers so $4$ divides one of them.
• If $3\nmid m$ then $3\mid m^2-1=(m-1)(m+1)$ because $3$ divides (exactly) one of the three consecutive integers $m-1,m,m+1$.

Now if $24\mid mn+1$, then $m$ is odd and not divisible by $3$, so $24=3\cdot8\mid m^2-1$ and $$24\mid m\times(mn+1)-n\times(m^2-1)=m+n.$$