# $15a ≡ ca \pmod{25}$, then $15 ≡ c \pmod{25}$

For which numbers $a$ is it true that if $15a ≡ ca \pmod{25}$, then $15 ≡ c \pmod{25}$?

I know that this means that $a\frac{15-c}{25}=k_1\in \mathbb{Z}$ and $\frac{15-c}{25}=k_2\in \mathbb{Z}$, but what must I show?

• I changed (\text{mod}~25) to \pmod{25}. That is standard. It automatically generated proper spacing before and after "mod" and puts the parentheses where they should be. – Michael Hardy Apr 29 '13 at 23:04

Recall that if $x \mid yz$ and $\text{gcd}(x,y)=1$, i.e., $x$ and $y$ are relatively prime, then $x \mid z$.
• @Pinsgrair $\gcd(x,y) = 1$. – user17762 Apr 29 '13 at 23:03
• @Pinsgrair: "relatively prime" means "$\gcd(x, y) = 1$". "$(x, y)$" above is another notation for the gcd. – The_Sympathizer Apr 29 '13 at 23:04
• @user17762 What are your $x$ and $y$ here? – Pinsgrair Apr 29 '13 at 23:10