If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$? [duplicate]
If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
when, $a=3$
we have, $a+2=5$ and $a+4=7$
Solution 1:
HINT: One of the numbers $a,a+2$, and $a+4$ must be divisible by $3$. Why?
Solution 2:
$a\equiv 0 \mod 3\Rightarrow a=3$
$a\equiv 1\mod 3\Rightarrow a+2\equiv 0\mod 3\Rightarrow a+2=3\Rightarrow a=1$
$a\equiv 2\mod 3\Rightarrow a+4\equiv0\mod 3\Rightarrow a+4=3\Rightarrow a=-1$
So the only possibility is the first one.