In Section 3.4 we used the addition of hours, weekdays, and months as examples of modular arithmetic. Now we define binary operations on the sets \(\Z_n=\{0,1,2,\dots,n-1\}\text{,}\) where \(n\in\N\text{,}\) using integer addition and multiplication. For \(a\) and \(b\) in \(\Z_n\text{,}\) we use \((a+b)\fmod n\) and \((a\cdot b)\fmod n\text{.}\) The remainder after division by \(n\) is always in \(\Z_n=\{0,1,2,\dots,n-1\}\text{.}\) Therefore, these rules define binary operations on \(\Z_n\text{.}\)
We saw operation tables for modular addition and multiplication in Chapter 13. In Table 14.3.3 the tables for addition and multiplication modulo \(7\) appear side by side. Once these tables are available, we can use them to look up the result of modular addition or multiplication.
We present examples for addition and multiplication modulo 7. Let \(a\oplus b:=(a+b)\fmod 7\) and \(a\otimes b:=(a\cdot b)\fmod 7\text{.}\) Tables for the binary operations \(\oplus\) and \(\otimes\) are given in Table 14.3.3.
Table14.3.3.Addition and multiplication tables for arithmetic modulo 7, that is, for the operations given by \(a\oplus b=(a+b)\fmod 7\) and \(a\otimes b=(a\cdot b)\fmod 7\text{.}\)
In the next two sections, we use modular addition and multiplication to define certain groups. For any \(n\in\N\text{,}\) the set \(\Z_n\) with addition modulo \(n\) is a group. For any prime number \(p\text{,}\) the set \(\Z_p^\otimes\) with multiplication modulo \(p\) is also a group.
