Consecutive Integers Range Determination

The problem states we are dealing with 7 consecutive integers. Let these integers be represented as $n, n+1, n+2, n+3, n+4, n+5, n+6$. We are given two conditions regarding the smallest and greatest of these integers:

• The smallest integer is greater than 60. This means $n > 60$.
• The greatest integer is less than 70. This means $n+6 < 70$, which simplifies to $n < 64$.

Combining these inequalities, we find that $60 < n < 64$. The possible integer values for $n$ are 61, 62, and 63.

Let's examine the sets of 7 consecutive integers for each possible value of $n$:

• If $n=61$, the set is $\{61, 62, 63, 64, 65, 66, 67\}$. The smallest is 61 (which is > 60) and the greatest is 67 (which is < 70). This set is valid.
• If $n=62$, the set is $\{62, 63, 64, 65, 66, 67, 68\}$. The smallest is 62 (> 60) and the greatest is 68 (< 70). This set is also valid.
• If $n=63$, the set is $\{63, 64, 65, 66, 67, 68, 69\}$. The smallest is 63 (> 60) and the greatest is 69 (< 70). This set is also valid.

Prime Number Clue Assignment

Clue (I) states that A and B are both prime numbers. We need to find which of the possible sets contains exactly two prime numbers.

• In the set $\{61, 62, 63, 64, 65, 66, 67\}$, the prime numbers are 61 and 67. This set allows for two distinct prime numbers, A and B.
• In the set $\{62, 63, 64, 65, 66, 67, 68\}$, the only prime number is 67. This set does not allow for two distinct prime numbers A and B.
• In the set $\{63, 64, 65, 66, 67, 68, 69\}$, the only prime number is 67. This set also does not allow for two distinct prime numbers A and B.

Therefore, the correct set of 7 consecutive integers must be $\{61, 62, 63, 64, 65, 66, 67\}$. From this set, the prime numbers are 61 and 67. Thus, $\{A, B\} = \{61, 67\}$.

Multiple of 9 and Same Digit Clues

Let's use the other clues to identify the remaining variables:

• Clue (II): T is a multiple of 9. In the confirmed set $\{61, 62, 63, 64, 65, 66, 67\}$, the only multiple of 9 is 63. So, T = 63.
• Clue (III): Both the digits of P are same. In the set $\{61, 62, 63, 64, 65, 66, 67\}$, the number with identical digits is 66. So, P = 66.

Average and Difference Clues for R and S

Clue (IV): The average of R and S is 63 and the difference between R and S is 2.

We can set up equations based on this clue:

• The average of R and S is 63: $ \frac{R+S}{2} = 63 $ Multiplying both sides by 2 gives: $ R+S = 126 $
• The difference between R and S is 2. Assuming $R$ is the larger number, we have: $ R-S = 2 $

Now we solve this system of two linear equations:

1. $R+S = 126$
2. $R-S = 2$

Adding equation (1) and equation (2):

Substitute the value of $R$ back into equation (1):

So, R = 64 and S = 62. Both these numbers are present in our confirmed set $\{61, 62, 63, 64, 65, 66, 67\}$.

Identifying Q and A Value

We have assigned values to T, P, R, and S from the set $\{61, 62, 63, 64, 65, 66, 67\}$:

• T = 63
• P = 66
• R = 64
• S = 62
• A and B are the primes: 61 and 67.

The only integer remaining in the set that has not been assigned a variable is 65. The only variable left unassigned is Q. Therefore, Q = 65.

The question asks for the sum of A and Q, given that A is smaller than B. Since $\{A, B\} = \{61, 67\}$ and A must be smaller:

• A = 61

Final Sum Calculation

We need to calculate the sum of A and Q.

• A = 61
• Q = 65

The sum is:

The sum of A and Q is 126.