Question:medium

Five numbers A, B, C, D and E are to be arranged in an array in such a manner that they have a common prime factor between two consecutive numbers. These integers are such that: A has a prime factor P. B has two prime factors Q and R. C has two prime factors Q and S. D has two prime factors P and S. E has two prime factors P and R.

If number E is not in the list and the other four numbers are arranged properly, which of the following must be true?

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Work out which pairs among A, B, C, D actually share a prime factor, then see which two numbers only have one possible neighbor each.
Updated On: Jul 13, 2026
  • A and D can not be the consecutive numbers
  • A and B are to be placed at the two ends in the array
  • A and C are to be placed at the two ends in the array
  • C and D can not be the consecutive numbers
Show Solution

The Correct Option is B

Solution and Explanation

Once E is taken out of the picture, only A, B, C and D remain, with factors A: P, B: Q and R, C: Q and S, D: P and S. Since only pairs sharing a prime factor can sit together, work out which of the four can sit beside which.

  1. (A) A and D can not be the consecutive numbers: A and D actually share the factor P, so they can sit together, and in the only valid row they always do. This statement is false.
  2. (B) A and B are to be placed at the two ends in the array: A shares a factor with only D, through P, and B shares a factor with only C, through Q. A number with just one possible partner cannot be placed in a middle seat, so both A and B are forced to the two end seats, giving the row A, D, C, B or its reverse. This statement is true.
  3. (C) A and C are to be placed at the two ends in the array: A and C share no prime factor at all, and the valid row places B, not C, at the end opposite A. This statement is false.
  4. (D) C and D can not be the consecutive numbers: C and D share the factor S, so they do sit together in the valid row A, D, C, B. This statement is false.

Only option (B) matches the row that actually works, A, D, C, B or its mirror image B, C, D, A, where A and B occupy the two end positions.

Let's summarize:

  • A connects only to D, and B connects only to C, so both are stuck at the ends of the row.
  • The one valid arrangement is A, D, C, B, or its reverse.

So the statement that must be true is option (B): A and B are placed at the two ends of the array.

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