CAT 2024 Slot 2 QA Q1 Number Systems

Question 1 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

When \(3^{333}\) is divided by 11 , the remainder is
  1. A
    1
  2. B
    10
  3. C
    6
  4. D
    5

Answer: (D) 5 Number Systems MCQ

Solution

  1. \(3^5 = 243 = 22 \times 11 + 1\), so \(3^5 \equiv 1 \pmod{11}\).
  2. \(333 = 5 \times 66 + 3\), so \(3^{333} = (3^5)^{66} \cdot 3^3 \equiv 27 \pmod{11}\).
  3. \(27 = 2 \times 11 + 5\), so the remainder is 5.

Shortcut. Find the cycle length of powers modulo 11 (here 5) and reduce the exponent.

Same concept in other years

See every question on Remainder of a large power: find the cycle of remainders.

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