how to find the smallest positive integer which solves the following
q=5(mod17) and q=8(mod12)?

Question

how to find the smallest positive integer   which solves the following 
           q=5(mod17) and q=8(mod12)?

asnaseer · Answer

yo can write this as follows:$$q=17m_1+5=12m_2+8$$so you need to find the smallest values for $m_1$ and $m_2$. we can write $m_2$ in terms of $m_1$ as:$$m_2=(17m_1-3)/12$$ok, so lets try increasing values of $m_1$ starting with $m_1=1$ and see if we can find a suitable $m_2$:$$\begin{align}m_1&=1\implies m_2=14/12\quad&\text{reject}\
&=2\implies m_2=31/12\quad&\text{reject}\
&=3\implies m_2=48/12=4\quad&\text{bingo!}\
\end{align}$$
so $m_1=3$ gives us a viable solution, which gives us:$$q=17m_1+5=17*3+5=51+5=56$$