Which of the following statements are correct?
(A) \(83 \equiv 3 \pmod{8}\)
(B) \(-17 \equiv 3 \pmod{5}\)
(C) \(45 \equiv 12 \pmod{9}\)
(D) \(2^{10} \equiv 1 \pmod{11}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
By congruence modulo n, \(a \equiv b \pmod{n}\) means that \(a - b\) is a multiple of n.
\(83 - 3 = 80 = 8 \times 10\)
\(-17 - 3 = -20 = 5 \times (-4)\)
\(45 - 12 = 33\)
\(2^{10} - 1 = 1023 = 11 \times 93\)
So the differences are 80, \(-20\), 33 and 1023, and only 33 fails to be a multiple of its modulus 9.
Checking the statements
(A) \(83 \equiv 3 \pmod{8}\) — Correct. \(83 - 3 = 80\), a multiple of 8.
(B) \(-17 \equiv 3 \pmod{5}\) — Correct. \(-17 - 3 = -20\), a multiple of 5.
(C) \(45 \equiv 12 \pmod{9}\) — Incorrect. \(45 - 12 = 33\), not a multiple of 9 (45 leaves 0, 12 leaves 3).
(D) \(2^{10} \equiv 1 \pmod{11}\) — Correct. \(1024 - 1 = 1023 = 11 \times 93\).