Question 20

For a positive integer $$n$$, let $$n\bmod13$$ denote the remainder $$r$$, $$0\leq r<13$$ when divided by $$13$$. If $$a,b,c$$ are integers such that $$4a+5b+6c = 1 mod13 a-b-7c=3 mod13 3a-4b+5c=9 mod13$$, then $$a+b+c\bmod13$$ is $$\underline{\qquad}$$.

Correct Answer: 5

Solution

From the second congruence, $$a\equiv3+b+7c\pmod{13}$$. Substitution into the other two gives $$9b+8c\equiv2\pmod{13}$$ and $$b\equiv0\pmod{13}$$. Hence $$c\equiv10$$ and $$a\equiv8$$, so $$a+b+c\equiv18\equiv5\pmod{13}$$.

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For a positive integer $$n$$, let $$n\bmod13$$ denote the remainder $$r$$, $$0\leq r<13$$ when divided by $$13$$. If $$a,b,c$$ are integers such that $$4a+5b+6c = 1 mod13 a-b-7c=3 mod13 3a-4b+5c=9 mod13$$, then $$a+b+c\bmod13$$ is $$\underline{\qquad}$$.

Solution

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