Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$.

Question

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$.

I am very new to proofs and not completely sure of how to approach this one. I tried several different values for $x$ other than $5$ and came up with values that are not prime. However, I can't see how I could generalize this question to prove that it works if $x$ is $5$.

Help would be appreciated.

Thanks :)

Answer 1

Hint: Show that one of the numbers is a multiple of $5$. One way to do that: Write $x=5k+r$.

Answer 2

HINT: try $x\equiv 0,1,2,3,4\mod 5$ it works only $x=5$

Answer 3

$x\bmod5=x\bmod5$
$(x+6)\bmod5=(x+1)\bmod5$
$(x+12)\bmod5=(x+2)\bmod5$
$(x+18)\bmod5=(x+3)\bmod5$
$(x+24)\bmod5=(x+4)\bmod5$

so if $x\ne5$, then $5$ must divide one of the five integers, and it can't be $5$ itself, whence it must be composite.