Beginning Activity 1: Congruence Modulo 6

Beginning Activity Beginning Activity 1: Congruence Modulo 6
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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 $a \equiv 3 (\mod 6)$ and find five different integers $b$ such that $b \equiv 4 (\mod 6) .$ That is, find five different integers in $[3],$ the congruence class of 3 modulo 6 and five different integers in $[4],$ 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 $0 \leq r < 6$ and $s \equiv r (\mod 6) .$ 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 $0 \leq r < 6$ and $p \equiv r (\mod 6) .$ 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 $0 \leq r < 6$ and $q \equiv r (\mod 6) .$ What do you observe?

Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

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Beginning Activity Beginning Activity 1: Congruence Modulo 6
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