Which of the following are correct?
(A) If \(a \equiv b \pmod{n}\), then \(-a \equiv -b \pmod{n}\)
(B) If \(a + b = c\), then \(a \pmod{n} + b \pmod{n} \equiv (a+b+c) \pmod{n}\).
(C) If \(a \equiv b \pmod{n}\), then \(ka \equiv kb \pmod{n},\ \forall\, k \in I\).
(D) If \(a \equiv b \pmod{n}\), then \((a+k) \equiv (b+k) \pmod{n},\ \forall\, k \in I\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
By definition, \(a \equiv b \pmod{n}\) means that n divides a − b. Here I is the set of integers.
Checking the statements
(A) If \(a \equiv b \pmod{n}\), then \(-a \equiv -b \pmod{n}\) — Correct. \((-a) - (-b) = -(a - b)\), and n divides it since n divides a − b.
(B) If \(a + b = c\), then \(a \pmod{n} + b \pmod{n} \equiv (a+b+c) \pmod{n}\). — Incorrect. The left side is congruent to \(a + b = c\), but the right side is \(a + b + c = 2c\). These agree only when n divides c. For example, with a = 1, b = 1, c = 2 and n = 5:
\( \displaystyle a + b = 2 \)
\( \displaystyle a + b + c = 4 \)
\( \displaystyle 2 \not\equiv 4 \pmod{5} \)
(C) If \(a \equiv b \pmod{n}\), then \(ka \equiv kb \pmod{n},\ \forall\, k \in I\). — Correct. \(ka - kb = k(a - b)\) is a multiple of a − b, so n divides it.
(D) If \(a \equiv b \pmod{n}\), then \((a+k) \equiv (b+k) \pmod{n},\ \forall\, k \in I\). — Correct. \((a + k) - (b + k) = a - b\), which n divides.