Let \(a\), \(b\) and \(c\) be integers and let \(m \ge 2\) be an integer. Which of the following statements are always true?
(A) If \(a \equiv b \pmod{m}\), then \(a + c \equiv b + c \pmod{m}\).
(B) If \(a \equiv b \pmod{m}\), then \(ac \equiv bc \pmod{m}\).
(C) If \(a \equiv b \pmod{m}\), then \(a^{2} \equiv b^{2} \pmod{m}\).
(D) If \(ac \equiv bc \pmod{m}\) and \(c \not\equiv 0 \pmod{m}\), then \(a \equiv b \pmod{m}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
By definition, \(a \equiv b \pmod{m}\) means that \(m\) divides \(a - b\). Each rule is checked by looking at the difference of the two sides.
\((a + c) - (b + c) = a - b\)
\(ac - bc = (a - b)c\)
\(a^{2} - b^{2} = (a - b)(a + b)\)
So the first three differences are multiples of \(a - b\), while cancelling \(c\) fails for \(m = 6\), \(a = 3\), \(b = 0\), \(c = 2\).
Checking the statements
(A) If \(a \equiv b \pmod{m}\), then \(a + c \equiv b + c \pmod{m}\). — Correct. The difference is \(a - b\) itself, a multiple of \(m\).
(B) If \(a \equiv b \pmod{m}\), then \(ac \equiv bc \pmod{m}\). — Correct. The difference \((a - b)c\) is a multiple of \(m\).
(C) If \(a \equiv b \pmod{m}\), then \(a^{2} \equiv b^{2} \pmod{m}\). — Correct. The difference \((a - b)(a + b)\) is a multiple of \(m\).
(D) If \(ac \equiv bc \pmod{m}\) and \(c \not\equiv 0 \pmod{m}\), then \(a \equiv b \pmod{m}\). — Incorrect. \(3 \times 2 \equiv 0 \times 2 \pmod{6}\) since both are multiples of 6, but \(3 \not\equiv 0 \pmod{6}\).