FORM GR1268 Question 23

Question:
Let \((\mathbb Z_{10}, +,\cdot)\) be the ring of integers modulo \(10\), and let \(S\) be the subset of \(\mathbb Z_{10}\) represented by \(\{0,2,4,6,8\}\). Which of the following statement is FALSE?

(A) \((S,+,\cdot)\) is closed under addition modulo \(10\).
(B) \((S,+,\cdot)\) is closed under multiplication modulo \(10\).
(C) \((S,+,\cdot)\) has an identity under addition modulo \(10\).
(D) \((S,+,\cdot)\) has no identity under multiplication modulo \(10\).
(E) \((S,+,\cdot)\) is commutative under addition modulo \(10\).

Answer:
(D)

Answer Key:
\(S\) contains all even remainders.

(A) is true because even + even = even, and even number has an even remainder when divided by \(10\).

(B) is true because even \(\cdot\) even = even, and even number has an even remainder when divided by \(10\).

(C) is true because \(0\) is additive identity.

(D) is not true because \(6\) is multiplicative identity.
That is,
\(0\cdot6=0\equiv0\pmod{10}\)
\(2\cdot6=12\equiv2\pmod{10}\)
\(4\cdot6=24\equiv4\pmod{10}\)
\(6\cdot6=36\equiv6\pmod{10}\)
\(8\cdot6=48\equiv8\pmod{10}\)

(E) is true addition of integers are commutative.