This problem asks us to find the sum of the digits of the number Q, given a set of 7 consecutive integers P, Q, R, S, T, A, and B. We are provided with specific conditions about these integers.

Determining the Set of Consecutive Integers

First, let's establish the range for these integers. We know the smallest integer is greater than 60, and the greatest is less than 70. Since they are consecutive, the possible sets of 7 integers are:

• {61, 62, 63, 64, 65, 66, 67}
• {62, 63, 64, 65, 66, 67, 68}
• {63, 64, 65, 66, 67, 68, 69}

We will use the given conditions to identify which set is correct and assign values to the variables.

Applying the Given Conditions

Condition (I): A and B are prime numbers

We need to find the prime numbers within the possible ranges. The prime numbers between 60 and 70 are 61 and 67.

Therefore, A and B must be 61 and 67 (in any order).

Condition (II): T is a multiple of 9

We look for a multiple of 9 within the range 61 to 69.

The only multiple of 9 in this range is 63.

Therefore, T = 63.

Condition (III): Both digits of P are the same

We need to find a number within the potential range where both digits are identical.

The only number in the range [61, 69] with identical digits is 66.

Therefore, P = 66.

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

Let's use these two pieces of information:

• Average: $\frac{R + S}{2} = 63$
• Difference: $|R - S| = 2$

From the average, we get:

$R + S = 2 \times 63 = 126$

From the difference, we have two possibilities:

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

Case 1: $R - S = 2$. Adding this to $R + S = 126$:

$(R - S) + (R + S) = 2 + 126$

$2R = 128 \implies R = 64$

Substituting R back into $R + S = 126$:

$64 + S = 126 \implies S = 126 - 64 = 62$

So, R=64 and S=62.

Case 2: $S - R = 2$. Adding this to $R + S = 126$:

$(S - R) + (R + S) = 2 + 126$

$2S = 128 \implies S = 64$

Substituting S back into $R + S = 126$:

$R + 64 = 126 \implies R = 126 - 64 = 62$

So, R=62 and S=64.

In either case, R and S are 62 and 64.

Identifying the Value of Q

So far, we have identified the following numbers and their corresponding variables:

• A, B = {61, 67} (primes)
• T = 63 (multiple of 9)
• P = 66 (same digits)
• R, S = {62, 64} (average 63, difference 2)

The numbers identified are {61, 62, 63, 64, 66, 67}. These are 6 distinct numbers.

We need a set of 7 *consecutive* integers. Let's check which of the possible sets contains these 6 numbers.

• Set 1: {61, 62, 63, 64, 65, 66, 67}
• Set 2: {62, 63, 64, 65, 66, 67, 68}
• Set 3: {63, 64, 65, 66, 67, 68, 69}

Set 1, {61, 62, 63, 64, 65, 66, 67}, contains all the identified numbers {61, 62, 63, 64, 66, 67}.

The variables assigned are P, Q, R, S, T, A, B. The numbers identified cover all variables except Q.

The number missing from the identified list {61, 62, 63, 64, 66, 67} within Set 1 is 65.

Therefore, the remaining variable Q must be 65.

Calculating the Sum of the Digits of Q

We found that Q = 65.

The sum of the digits of Q is calculated as:

$6 + 5 = 11$

Thus, the sum of the digits of the number Q is 11.