Question

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 the number E is arranged in the middle with two numbers on either side of it, all of the following must be true, EXCEPT:

• A. A and D are arranged consecutively
• B. B and C are arranged consecutively
• C. B and E are arranged consecutively
• D. A is arranged at one end in the array

Hint:

Work out who can actually sit beside E (only A, B or D can, since C shares nothing with E), then list out every row that fits and test each option against all of them.

Answer 1

The Correct Option is D

Step 1: Note E's factors and work out who can sit next to it.
E has the prime factors P and R. A neighbor of E must share P or R with it. A has P, so A can sit next to E. D has P and S, so D can sit next to E. B has Q and R, so B can sit next to E. C has only Q and S, neither of which is P or R, so C can never sit directly next to E.

Step 2: Use this to fix where C must go.
E sits in the middle of the row of five, with two numbers on each side, so E's two direct neighbors must come from A, B and D. Since C cannot be one of E's neighbors, C must sit at one of the two ends of the row.

Step 3: Work out the neighbor of C.
C has factors Q and S. Its one neighbor, since C sits at an end, must share Q or S with it. Checking A (only P), B (Q, R) and D (P, S): B shares Q with C, and D shares S with C, but A shares nothing with C. So A can never sit right next to C either.

Step 4: Build out the complete valid rows.
Put C at the left end. Its neighbor must be B or D, not A. If C sits next to B, the row so far is C, B, E, _, _, and the two remaining seats need A and D. Both connect to E through P, and A and D also share P with each other, so both C, B, E, A, D and C, B, E, D, A work. If C sits next to D instead, the row so far is C, D, E, _, _, needing A and B in the last two seats, but A and B share no factor with each other, so neither order works there. With C on the left, the valid rows are C, B, E, A, D and C, B, E, D, A. By the same reasoning with C on the right end, the mirror rows D, A, E, B, C and A, D, E, B, C are also valid, giving four valid rows in total.

Step 5: Test each option against all four valid rows.
(A) A and D consecutive: true in all four rows, since A and D always sit side by side at one end.
(B) B and C consecutive: true in all four rows, since B always sits right beside C.
(C) B and E consecutive: true in all four rows, since B is always the number placed directly next to E.
(D) A is at one end: false in the row C, B, E, A, D, where A sits in the fourth seat, between D and E, not at either end.

Final Answer:
Statement (D) is not always true, so it is the exception.
\[ \boxed{\text{(D) A is arranged at one end in the array}} \]