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?
Which of the following is an acceptable order, from left to right, in which the numbers can be arranged?
Show Hint
- D, E, B, C, A
- B, A, E, D, C
- B, C, D, E, A
- B, C, E, D, A
The Correct Option is C
Solution and Explanation
Every number in the array must share a prime factor with the number right next to it. We are given: A has the factor P; B has factors Q and R; C has factors Q and S; D has factors P and S; E has factors P and R. Check each option pair by pair, from left to right.
- (A) D, E, B, C, A: D-E share P, E-B share R, B-C share Q, but C-A has no common factor at all (C has Q, S while A only has P). This order breaks at the last pair, so it does not work.
- (B) B, A, E, D, C: B-A already fails, since B has Q, R and A has only P, with nothing in common. This order does not work.
- (C) B, C, D, E, A: B-C share Q, C-D share S, D-E share P, E-A share P. Every adjacent pair shares a prime factor, so this order holds together all the way through.
- (D) B, C, E, D, A: B-C share Q, but C-E has no common factor (C has Q, S while E has P, R). This order breaks at the second pair.
Only option (C), B, C, D, E, A, keeps a common prime factor between every pair of consecutive numbers, so it is the acceptable order.
Let's summarize:
- A only connects through P, so it can only sit next to D or E.
- C only connects through Q or S, so it can only sit next to B or D.
- Checking each option pair by pair from left to right quickly rules out the broken chains.
The order B, C, D, E, A is the only one where every pair of consecutive numbers shares a common prime factor.
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