Beginning Activity Beginning Activity 1: Congruence Modulo 6
For this activity, we will only use the relation of congruence modulo 6 on the set of integers.
1.
Find five different integers \(a\) such that \(a \equiv 3 \pmod 6\) and find five different integers \(b\) such that \(b \equiv 4 \pmod 6\text{.}\) That is, find five different integers in \([3]\text{,}\) the congruence class of 3 modulo 6 and five different integers in \([4]\text{,}\) the congruence class of 4 modulo 6.
2.
Calculate \(s = a + b\) using several values of \(a\) in \([3]\) and several values of \(b\) in \([4]\) from Exercise 1. For each sum \(s\) that is calculated, find \(r\) so that \(0 \leq r \lt 6\) and \(s \equiv r \pmod 6\text{.}\) What do you observe?
3.
Calculate \(p = a \cdot b\) using several values of \(a\) in \([3]\) and several values of \(b\) in \([4]\) from Exercise 1. For each product \(p\) that is calculated, find \(r\) so that \(0 \leq r \lt 6\) and \(p \equiv r \pmod 6\text{.}\) What do you observe?
4.
Calculate \(q = a^2\) using several values of \(a\) in \([3]\) from Exercise 1. For each product \(q\) that is calculated, find \(r\) so that \(0 \leq r \lt 6\) and \(q \equiv r \pmod 6\text{.}\) What do you observe?