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Determine the sum of all prime numbers \(p\) for which there exists no integer solution in $x$ to the congruence \(3(6x+1)\equiv 4\pmod p\).

 

 

Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$.

 Dec 21, 2018
 #1
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\(3(6x+1) \equiv 4 \pmod{p}\\ 18x + 3 \equiv 4 \pmod{p}\\ 18x \equiv 1 \pmod{p}\\ 3^3 \cdot 2 x \equiv 1 \mod{p}\\ \text{It should be clear any integer }x \text{ will cause }\\ 18x \equiv 0 \pmod{2} \text{ and }18x \equiv 0 \pmod{3} \\ \text{there will be an integer solution for all other primes}\\ 2+3=5\)

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 Dec 21, 2018

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