If $5x =7\pmod{13}$ and $3y =10\pmod{13}$. What is the value of $k$ in $xy =k\pmod{13}$
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Hint $\rm\,\ mod\ 13\!:\,\ \color{#C00}7\equiv \color{#0A0}{-6}\ \Rightarrow\ x\cdot y\, \equiv\, \dfrac{\color{#C00}7}{5}\cdot\dfrac{10}3\, \equiv\, \dfrac{\color{#0A0}{-6}}5\cdot \dfrac{10}3\,\equiv\, -2\cdot 2$ |
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We have that $(5x)(3y)=5 \pmod{13}$. As well $5\cdot 3=2\pmod{13}$. So we just need to find the inverse of $2$ under mod 13. This is easily seen to be 7. Hence, $xy=5\cdot 7\pmod{13}=9\pmod{13}$ |
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