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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?

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  • $\begingroup$ 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. $\endgroup$ – Michael Hardy Apr 29 '13 at 23:04
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HINT

Recall that if $x \mid yz$ and $\text{gcd}(x,y)=1$, i.e., $x$ and $y$ are relatively prime, then $x \mid z$.

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  • $\begingroup$ To be "relatively prime" is to be what? $\endgroup$ – Pinsgrair Apr 29 '13 at 23:01
  • $\begingroup$ @Pinsgrair $\gcd(x,y) = 1$. $\endgroup$ – user17762 Apr 29 '13 at 23:03
  • $\begingroup$ @Pinsgrair: "relatively prime" means "$\gcd(x, y) = 1$". "$(x, y)$" above is another notation for the gcd. $\endgroup$ – The_Sympathizer Apr 29 '13 at 23:04
  • $\begingroup$ You mean "coprime," yes? $\endgroup$ – Pinsgrair Apr 29 '13 at 23:06
  • $\begingroup$ @user17762 What are your $x$ and $y$ here? $\endgroup$ – Pinsgrair Apr 29 '13 at 23:10

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