Question:hard

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.

Which of the following is an acceptable order, from left to right, in which the numbers can be arranged?

Show Hint

Check each option pair by pair, left to right, and see whether the two numbers in every adjacent pair actually share a letter (P, Q, R or S).
Updated On: Jul 13, 2026
  • D, E, B, C, A
  • B, A, E, D, C
  • B, C, D, E, A
  • B, C, E, D, A
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Turn the clues into a factor list.
A: P. B: Q, R. C: Q, S. D: P, S. E: P, R. Two numbers are allowed to sit next to each other only if their factor lists share a letter.

Step 2: Build the list of allowed neighbor pairs.
Checking every pair of the five numbers: A pairs with D (common P) and with E (common P). B pairs with C (common Q) and with E (common R). C pairs with D (common S). D pairs with E (common P). No other pair shares a letter, so in particular A cannot sit next to B or C, and B cannot sit next to D.

Step 3: Check each option against this allowed-pairs list.
(A) D, E, B, C, A needs the pair C-A at the end, which is not in the allowed list, so it fails.
(B) B, A, E, D, C needs the pair B-A right at the start, which is not allowed, so it fails immediately.
(C) B, C, D, E, A needs the pairs B-C, C-D, D-E and E-A, and every one of these is in the allowed list.
(D) B, C, E, D, A needs the pair C-E, which is not allowed, so it fails at the second step.

Step 4: Pick the order that survives every check.
Only option (C) uses allowed pairs all the way through, so it is the order that works.

Final Answer:
The acceptable order is B, C, D, E, A.
\[ \boxed{\text{B, C, D, E, A}} \]
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