UPSC Prelims 2022 CSAT Paper II · Q78 of 78 Basic Numeracy easy

The sum of three consecutive integers is equal to their product. How many such possibilities are there?

  1. (a) Only one
  2. (b) Only two
  3. (c) Only three ✓ UPSC's answer
  4. (d) No such possibility is there

Why the answer is (c)

• Let the three consecutive integers be n-1, n, and n+1.

• Their sum is 3n, and their product is n(n-1)(n+1) = n(n^2 - 1).

• Equating sum and product gives 3n = n(n^2 - 1), so n(n^2 - 4) = 0.

• Therefore n = 0, n = 2, or n = -2.

• These give the triples (-1, 0, 1), (1, 2, 3), and (-3, -2, -1), so there are exactly three possibilities, matching option (c).

Why the other options are wrong

(a) Only one
Only one is incorrect because the equation gives three valid middle values: 0, 2, and -2.
(b) Only two
Only two is incorrect because all three solutions n = 0, n = 2, and n = -2 satisfy the condition.
(d) No such possibility is there
No such possibility is incorrect because (1, 2, 3), (-1, 0, 1), and (-3, -2, -1) are valid examples.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.

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