Question

Recall that a|b (a divides b), if there exists an integer c, such that b = a · c.

(a) If a|b and a|c, show that a2|(b · c).
(b) Let a be an integer and d is a positive integer. Then there are unique integers q and r such
that 0  r (c) Write at least 3 positive integers and 3 negative integers in the equivalence class [12], given
m = 7.
(d) Find 128 mod 7 using equivalence classes. Show your work.

Answer 1

To solve the mathematics problems, we showed that if a divides both b and c, then a^2 divides b · c; identified six integers within the equivalence class of 12 modulo 7, including both positive and negative numbers; and calculated 128 mod 7 using equivalence classes to find the remainder is 3.

Step-by-step explanation:

Regarding the mathematics problems at hand:

(a) Proving that a^2 divides b · c

If a divides b and a divides c, there exist integers such that b = a · x and c = a · y. Multiplying both equations we get b · c = a^2 · x · y, proving that a^2 divides b · c.

(c) Positive and Negative Integers in Equivalence Class

For the equivalence class [12], given m = 7, positive integers can be 12, 19, 26 and negative integers can be 5, -2, -9 as each differs from 12 by 7n for some n.

(d) Finding 128 mod 7 Using Equivalence Classes

128 is equivalent to 3 modulo 7 as 128 = 7 · 18 + 3 and hence 128 mod 7 = 3.