benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / If a,b,c are positive integers less than 13 such that …a+b+c is divided by 13.

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

If a,b,ca,b,c are positive integers less than 1313 such that \begin{align*} 2ab+bc+ca&\equiv 0\pmod{13}\\ ab+2bc+ca&\equiv 6abc\pmod{13}\\ ab+bc+2ca&\equiv 8abc\pmod {13} \end{align*}then determine the remainder when a+b+ca+b+c is divided by 1313.
Plain-text mathematical notation (without MathML)
If a,b,c are positive integers less than 13 such that \begin{align*}
2ab+bc+ca&\equiv 0\pmod{13}\\
ab+2bc+ca&\equiv 6abc\pmod{13}\\
ab+bc+2ca&\equiv 8abc\pmod {13}
\end{align*}then determine the remainder when a+b+c is divided by 13.
Original LaTeX notation
If $a,b,c$ are positive integers less than $13$ such that \begin{align*}
2ab+bc+ca&\equiv 0\pmod{13}\\
ab+2bc+ca&\equiv 6abc\pmod{13}\\
ab+bc+2ca&\equiv 8abc\pmod {13}
\end{align*}then determine the remainder when $a+b+c$ is divided by $13$.

level

Level 5

type

Number Theory

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
93cef7c1-b104-5c60-ba38-f7b8aacd99c5
task_key
number~5ftheory--test--93cef7c1-b104-5c60-ba38-f7b8aacd99c5
task_revision_id
1
config / split
number_theory / test

Official source

initial import