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2. Recall that, for integers a and b, a ? b (mod 5) means that there is an integer k such that a ? b = 5k. (a) For each of the following congruences, find the integer k as per the above definition. i. 7 ? 7 (mod 5) ii. 7 ? 2 (mod 5) iii. 2 ? 7 (mod 5) iv. 12 ? 7 (mod 5) v. 12 ? 2 (mod 5) (b) Show that congruence modulo 5 is reflexive: that is, show that for every integer x, x ? x (mod 5). (Hint: look at 7 ? 7 (mod 5)) (c) Show that congruence modulo 5 is symmetric: that is, show that for all integers x and y, if x ? y (mod 5), then y ? x (mod 5). Hint: Look at 7 ? 2 (mod 5) and 2 ? 7 (mod 5). (d) Show that congruence modulo 5 is transitive: that is, show that for all integers x, y, and z, if x ? y and y ? z, then x ? z. Hint: look at 12 ? 7 (mod 5), 7 ? 2 (mod 5) and 12 ? 2 (mod 5) (e) Recall that a congruence class is a set of all numbers that are congruent to each other. Consider the congruence class A = {x ? Z : x ? 3 (mod 5)}. i. List 2 positive and 2 negative elements of A. ii. Are all elements of this congruence class also congruent modulo 10? Explain why or give a counterexample.

          2. Recall that, for integers a and b, a ? b (mod 5) means that there is an integer k such that a ? b = 5k.
(a) For each of the following congruences, find the integer k as per the above definition.
i. 7 ? 7 (mod 5)
ii. 7 ? 2 (mod 5)
iii. 2 ? 7 (mod 5)
iv. 12 ? 7 (mod 5)
v. 12 ? 2 (mod 5)
(b) Show that congruence modulo 5 is reflexive: that is, show that for every integer x, x ? x (mod 5). (Hint: look at 7 ? 7 (mod 5))
(c) Show that congruence modulo 5 is symmetric: that is, show that for all integers x and y, if x ? y (mod 5), then y ? x (mod 5). Hint: Look at 7 ? 2 (mod 5) and 2 ? 7 (mod 5).
(d) Show that congruence modulo 5 is transitive: that is, show that for all integers x, y, and z, if x ? y and y ? z, then x ? z. Hint: look at 12 ? 7 (mod 5), 7 ? 2 (mod 5) and 12 ? 2 (mod 5)
(e) Recall that a congruence class is a set of all numbers that are congruent to each other. Consider the congruence class A = {x ? Z : x ? 3 (mod 5)}.
i. List 2 positive and 2 negative elements of A.
ii. Are all elements of this congruence class also congruent modulo 10? Explain why or give a counterexample.

2. Recall that, for integers a and b, a ? b (mod 5) means that there is an integer k such that a ? b = 5k.
(a) For each of the following congruences, find the integer k as per the above definition.
i. 7 ? 7 (mod 5)
ii. 7 ? 2 (mod 5)
iii. 2 ? 7 (mod 5)
iv. 12 ? 7 (mod 5)
v. 12 ? 2 (mod 5)
(b) Show that congruence modulo 5 is reflexive: that is, show that for every integer x, x ? x (mod 5). (Hint: look at 7 ? 7 (mod 5))
(c) Show that congruence modulo 5 is symmetric: that is, show that for all integers x and y, if x ? y (mod 5), then y ? x (mod 5). Hint: Look at 7 ? 2 (mod 5) and 2 ? 7 (mod 5).
(d) Show that congruence modulo 5 is transitive: that is, show that for all integers x, y, and z, if x ? y and y ? z, then x ? z. Hint: look at 12 ? 7 (mod 5), 7 ? 2 (mod 5) and 12 ? 2 (mod 5)
(e) Recall that a congruence class is a set of all numbers that are congruent to each other. Consider the congruence class A = x ? Z : x ? 3 (mod 5).
i. List 2 positive and 2 negative elements of A.
ii. Are all elements of this congruence class also congruent modulo 10? Explain why or give a counterexample.

Added by Gonzalo C.

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