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How do you solve this congruence equation?
$$3\equiv -4a\pmod {13}$$
What I did was :
Applying symmetry property
$$ -4a\equiv 3\pmod {13} $$ Since gcd(13,4) = 1 we multiply both sides by inverse of $4\pmod {13}$
$$-a\equiv30\pmod{13}$$ $$a\equiv-30\pmod{13}$$ How can I continue from this point? $ a + 30 = 13k$ doesn't help me.

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  • $\begingroup$ You are almost done, you just need to simplify $-30\pmod{13}$. $\endgroup$
    – cansomeonehelpmeout
    Commented Jun 3, 2018 at 10:19
  • $\begingroup$ You could directly use the inverse of $-4$, which is $3$. $\endgroup$
    – Bernard
    Commented Jun 3, 2018 at 10:20
  • $\begingroup$ $-30\equiv -17...\bmod 13$. just keep going until you get a number in between $0$ and $12$. $\endgroup$
    – thesmallprint
    Commented Jun 3, 2018 at 10:23

2 Answers 2

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What I didn't know was that $-30\pmod{13} =9 $ is not equal to $30\pmod{13}=4$
So we can write the above expression as $$ a\equiv9\pmod{13}$$ $$ a = 9$$

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more handy solution:
3 = -4a (mod 13)
9 = -12a (mod 13)
9 = a (mod 13) <--i think you could figure it out.

so , a = 9, 21,......

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