Proving if $x$ divides $x+5$ then $x$ divides $5$

Question

Can someone show me how to prove if $x$ divides $x+5$ then $x$ divides $5$.

For a direct proof would you have to sub out the $5$ for some integer? Why?

Are there alternate ways to do this? Maybe induction?

Answer 1

Result: If $a|b$ and $a|c$ then $a|(b-c)$.

Since $x|(x+5)$ and $x|x$, we get $$x|[(x+5)-x]$$ and so $x|5$.

Answer 2

Hint. Write $5$ as the difference of two quantities both of which are divisible by $x$.

Answer 3

Write out the definitions:

$$xk=x+5\iff x(k-1)=5$$

Hence $x|5$.

Answer 4

Use this ,When $ a|b ,a|c \to a|mb+nc$ $$x|x+5\\x|x\\x|m(x+5)+n(x)$$ take $m=1,n=-1$ so $$x|1(x+5)-(x) \to x|5$$