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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You are almost done, you just need to simplify $-30\pmod{13}$. – cansomeonehelpmeout Jun 03 '18 at 10:19
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You could directly use the inverse of $-4$, which is $3$. – Bernard Jun 03 '18 at 10:20
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$-30\equiv -17...\bmod 13$. just keep going until you get a number in between $0$ and $12$. – thesmallprint Jun 03 '18 at 10:23
2 Answers
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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,......
JackalChan
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