Use the Chinese Remainder Theorem to find all solutions to the system of congruences
$$x\equiv 1\pmod 3$$ $$x\equiv3\pmod 5$$ $$x\equiv5\pmod 7$$
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8$$x\equiv-2\pmod{[3,5,7]}$$ – lab bhattacharjee Dec 25 '19 at 16:32
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See answers to these closely related questions chinese-remainder-theorem-modular , the-chinese-remainder-theorem ,resolve-a-system-using-chinese-remainder-theorem. – The Demonix _ Hermit Dec 25 '19 at 16:38
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Here's a subquestion when can't you use constant case ? – Dec 25 '19 at 16:55
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Welcome to MSE. Please don't just post homework questions with no effort shown. Show us what you tried and ask a specific question where you got stuck. – Ted Shifrin Dec 25 '19 at 17:36
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We have $$x+2\equiv0\pmod3$$ $$x+2\equiv0\pmod5$$ $$x+2\equiv0\pmod7$$ A number if divisible by $3$, $5$, $7$ iff it’s divisible by $3\times5\times7=105$. Therefore, we must have $$x+2\in\{105k\mid k\in\mathbb Z\}\Rightarrow$$ $$x\in\{105k-2\mid k\in\mathbb Z\}.$$
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