Question

Let n>0 be an integer. Let a∈Z, define [a]n​={a+nk∣k∈Z}. Use the definition given here to prove the following: if there exist a,a′,b,b′∈Z such that [a]n​=[a′]n​ and [b]n​=[b′]n​ then [a+b]n​=[a′+b′]n​ and [ab]n​=[a′b′]n​.

Answer 1

This response confirms that in modular arithmetic, if two pairs of integers are congruent modulo a positive integer n, then both their sums and products will also be congruent modulo n.

Step-by-step explanation:

The question concerns the properties of congruence classes in modular arithmetic. Using the definition [a]n = a + nk , where a, a', b, and b' are integers and n is a positive integer, we need to demonstrate that if [a]n = [a']n and [b]n = [b']n, then [a+b]n = [a'+b']n and [ab]n = [a'b']n.

To start, since a is congruent to a' modulo n and b is congruent to b' modulo n, it follows that there exist integers k and j such that a' = a + nk and b' = b + nj. For addition, we have a' + b' = (a + nk) + (b + nj) = (a + b) + n(k+j), which is in the set [a+b]n. For multiplication, we get a'b' = (a + nk)(b + nj) = ab + nbj + nak + n2kj = ab + n(bj + ak + nkj), which means a'b' is in the set [ab]n. Thus, the properties are proven.