Question

P, Q, R, S, T, A and B are consecutive integers (not necessarily in that order), such that the smallest of these is greater than 60 and the greatest is less than 70. It is known that:
(I) A and B both are prime numbers.
(II) T is a multiple of 9.
(III) Both the digits of P are same.
(IV) The average of R and S is 63 and the difference between R and S is 2.
What is the sum of A and Q if A is smaller than B?

• A. 126
• B. 128
• C. 120
• D. 122

Hint:

In logic puzzles with multiple constraints, identify the most powerful clues first. The clue about the two prime numbers was key to defining the exact set of integers, making the other clues much easier to apply.

Answer 1

The Correct Option is A

Step 1: Identify the set of seven consecutive integers.

- The integers are between 60 and 70 (exclusive), ranging from 61 to 69.
- Clue (I) states that A and B are prime. The primes in this range are 61 and 67. 
- To include both 61 and 67, the set must be \{61, 62, 63, 64, 65, 66, 67\}. 
- From Clue (I) and the condition "A is smaller than B", A = 61 and B = 67.

Step 2: Use the remaining clues.

- Clue (IV): The average of R and S is 63, thus \(R+S = 126\). Their difference is 2, so \(R-S=2\). Solving yields R = 64 and S = 62. Both are in the set. 
- Clue (III): P has identical digits. In our set, this is 66. So, P = 66. 
- Clue (II): T is a multiple of 9. In the set, this is 63. So, T = 63.

Step 3: Identify Q.

- Identified numbers: A=61, B=67, R=64, S=62, P=66, and T=63. 
- The remaining integer in \{61, 62, 63, 64, 65, 66, 67\} is 65. 
- Therefore, Q = 65.

Step 4: Calculate the sum.

- Find the sum of A and Q. 
\[ \text{Sum} = A + Q = 61 + 65 = 126 \]

Final Answer:

\[ \boxed{126} \]