+0

0
224
2
+30

If $a\equiv 18\pmod{42}$ and $b\equiv 73\pmod{42}$, then for what integer $n$ in the set $\{100,101,102,\ldots,140,141\}$ is it true that $$a-b\equiv n\pmod{42}~?$$

Apr 24, 2019

#1
+1

Apr 25, 2019
#2
+23903
+2

$$\begin{array}{|rcll|} \hline a &\equiv& 18\pmod{42} \\ b &\equiv& 73\pmod{42} \\ \hline a-b &\equiv& 18- 73 \pmod{42} \\ a-b &\equiv& -55 \pmod{42} \\ a-b &\equiv& -55+42+42 \pmod{42} \\ a-b &\equiv& 29 \pmod{42} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline a-b &\equiv& 29 \pmod{42} \\ a-b &=& 29 + z\cdot 42,\ z \in \mathbb{Z} \\ && \begin{array}{|r|c|c|} \hline & \mathbf{n} = (a-b)\pmod{42} & \\ z & =29 + z\cdot 42 & n \text{ in set} \\ \hline 0 & n=29 & \\ \hline 1 & n=71 & \\ \hline 2 & n=113 & \checkmark \\ \hline 3 & n=155 & \\ \hline \ldots \\ \hline \end{array} \\ \hline \end{array}$$

It is true for $$n = 113$$

Apr 25, 2019