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# Number Theory

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If $3x+7 \equiv 11$ (mod $16$), then 2x+11 is congruent (mod 16) to what integer between 0 and 15, inclusive?

Jul 11, 2021

### 1+0 Answers

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If $$3x+7 \equiv 11 \pmod{16}$$,
then 2x+11 is congruent $$\pmod{16}$$ to what integer between 0 and 15, inclusive?

$$\begin{array}{|rcll|} \hline 3x+7 &\equiv& 11 \pmod{16} \quad | \quad -7 \\ 3x &\equiv& 11-7 \pmod{16} \\ 3x &\equiv& 4 \pmod{16}\quad | \quad :3 \\ x &\equiv& \dfrac{4}{3} \pmod{16} \\ x \pmod{16} &\equiv& \dfrac{4}{3} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 2x+11\pmod{16} &\equiv& 2*\dfrac{4}{3}+11 \pmod{16} \\ &\equiv& \dfrac{8}{3}+11 \pmod{16} \\ &\equiv& \dfrac{8}{3}+\dfrac{33}{3} \pmod{16} \\ &\equiv& \dfrac{41}{3}\pmod{16} \quad | \quad 41\equiv 9 \pmod{16} \\ &\equiv& \dfrac{9}{3}\pmod{16} \\ \mathbf{2x+11\pmod{16}} &\equiv& \mathbf{3 \pmod{16}} \\ \hline \end{array}$$

Jul 12, 2021