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

• A. A and D can not be the consecutive numbers
• B. A and B are to be placed at the two ends in the array
• C. A and C are to be placed at the two ends in the array
• D. C and D can not be the consecutive numbers

Hint:

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.

Answer 1

The Correct Option is B

Step 1: Drop E and list the remaining factors.
With E removed, we are left with A (factor P), B (factors Q, R), C (factors Q, S) and D (factors P, S). Two of these can sit next to each other only if they share a prime factor.

Step 2: Work out every allowed neighbor pair among the four.
A and B: P versus Q, R, nothing common.
A and C: P versus Q, S, nothing common.
A and D: P versus P, S, they share P.
B and C: Q, R versus Q, S, they share Q.
B and D: Q, R versus P, S, nothing common.
C and D: Q, S versus P, S, they share S.
So the only allowed pairs are A-D, B-C and C-D.

Step 3: Notice that A and B each have only one possible neighbor.
A can only ever sit beside D, its one allowed partner, and B can only ever sit beside C. A number that can connect to just one other number cannot sit in the middle of the row, since a middle seat needs two neighbors, so A and B are forced to occupy the two end seats of the row of four.

Step 4: Build the row and check it.
Start from one end with A. Its only neighbor is D, so the row begins A, D. From D, the only unused allowed neighbor is C, since D-C share S, giving A, D, C. From C, the only unused allowed neighbor is B, since C-B share Q, completing the row A, D, C, B. Check every pair: A-D share P, D-C share S, C-B share Q. Every adjacent pair works, and this row (or its reverse, B, C, D, A) is the only way to arrange all four.

Step 5: Match this to the options.
In this row, A and B sit at the two ends, exactly as option (B) states. Option (A) is false since A and D are in fact always consecutive here. Option (C) is false since C sits in an inner seat, not at an end. Option (D) is false since C and D are consecutive in this row.

Final Answer:
A and B are always placed at the two ends of the array.
\[ \boxed{\text{(B) A and B are placed at the two ends}} \]