CAT 2024 Slot 1 QA Q10 Number Systems

Question 10 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

When \(10^{100}\) is divided by 7 , the remainder is
  1. A
    3
  2. B
    6
  3. C
    1
  4. D
    4

Answer: (D) 4 Number Systems MCQ

Solution

  1. \(10 \equiv 3 \pmod 7\), so we need \(3^{100} \bmod 7\).
  2. \(3^6 \equiv 1 \pmod 7\) (Fermat), and \(100 = 6 \times 16 + 4\).
  3. \(3^{100} \equiv 3^4 = 81 \equiv 4 \pmod 7\).

Shortcut. Powers of 10 mod 7 cycle 3, 2, 6, 4, 5, 1 with period 6; \(100 \bmod 6 = 4\) picks the 4th term: 4.

Same concept in other years

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

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