CUET · MATHS · PYQ PAPER 2025

Find the least non-negative remainder obtained when \(3^{128}\) is divided by \(7\) .

A \(2\)
B \(3\)
C \(4\)
D \(5\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(2\)

Step-by-step Solution

Detailed explanation

\(3^2 \equiv 2 \pmod{7}\) \(3^6 \equiv (3^2)^3 \equiv 2^3 \equiv 8 \equiv 1 \pmod{7}\) \(128 = 6 \times 21 + 2\) \(3^{128} \equiv (3^6)^{21} \cdot 3^2 \pmod{7}\) \(3^{128} \equiv 1^{21} \cdot 2 \pmod{7}\) \(3^{128} \equiv 2 \pmod{7}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Match List I with List - II. -

List - l	List - II
(A) Francis Crick	(l) DNA is the genetic material
(B) Watson and Crick	(II) Transforming principle
(C) Hershey and Chase	(III) Central dogma in molecular biology
(D) Frederick Griffith	(IV) Double helical structure of DNA

Choose the correct answer from the options given below:

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