CDS 2026 (I) Elementary Mathematics, question 21

Topic: Number system

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Q21

2727−940−37927^{27}-9^{40}-3^{79} is divisible by how many natural numbers less than 1010 ?

  1. (a) 11
  2. (b) 22
  3. (c) 33
  4. (d) More than 3Correct answer

Solution.

Answer: (d) More than 3

  1. Rewrite the number as 381−380−379=379(9−3−1)=5×3793^{81} - 3^{80} - 3^{79} = 3^{79}(9 - 3 - 1) = 5 \times 3^{79}.
  2. Among 1 through 9, its divisors are 1, 3, 5 and 9: four numbers.
  3. Do not omit 1 when counting natural-number divisors.