UPSC Prelims 2023 CSAT Paper II · Q17 of 77 Basic Numeracy medium

For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by :

  1. (a) 7 and 11 only
  2. (b) 11 and 13 only
  3. (c) 7 and 13 only
  4. (d) 7, 11 and 13 ✓ UPSC's answer

Why the answer is (d)

• A 6-digit number of the form XYZXYZ can be expressed as $1000 \times (100X + 10Y + Z) + (100X + 10Y + Z)$.

• This simplifies to $(100X + 10Y + Z) \times (1000 + 1)$, which equals $(100X + 10Y + Z) \times 1001$.

• The number 1001 can be factorized into prime factors as $7 \times 11 \times 13$.

• Since the number is a multiple of 1001, it is always divisible by 7, 11, and 13 regardless of the values of X, Y, and Z.

• Therefore, the number is divisible by 7, 11, and 13.

Why the other options are wrong

(a) 7 and 11 only
Option (a) is incorrect because it omits 13, which is also a factor of 1001.
(b) 11 and 13 only
Option (b) is incorrect because it omits 7, which is also a factor of 1001.
(c) 7 and 13 only
Option (c) is incorrect because it omits 11, which is also a factor of 1001.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.

Reading the answer is not the same as getting it right under a clock. Practise this question with UPSC's negative marking, and anything you miss goes into an error notebook until you get it right twice.

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