Study the following information carefully and answer the questions given below:

Eleven persons K, L, M, N, O, P, Q, R, S, T and U were born on the same day of the same month, (but not necessarily in the same order). All of them were born in different years—1981, 1983, 1987, 1994, 1997, 1999, 2000, 2003, 2006, 2007 and 2011 (but not necessarily in the same order). Their ages are to be calculated by considering the base year as 2023.

Age of L is a prime number. The age difference between L and Q is of 3 years. P is three persons younger than Q. O is either five persons elder than L or five persons younger than L. The number of persons elder to O is one less than the number of persons younger to N. T is immediately younger to N. The age difference between R and T is 13 years. R is not immediately elder to Q. Age of M is completely divisible by 4 and is elder to K. S is 2 years elder to K.

What is the product of ages of U and Q?

To find the product of the ages of U and Q, we need to determine the ages of all eleven persons based on the given clues.

First, let's calculate the age of each person in the base year 2023 based on their birth years:
- 1981: 2023 - 1981 = 42 years
- 1983: 2023 - 1983 = 40 years
- 1987: 2023 - 1987 = 36 years
- 1994: 2023 - 1994 = 29 years
- 1997: 2023 - 1997 = 26 years
- 1999: 2023 - 1999 = 24 years
- 2000: 2023 - 2000 = 23 years
- 2003: 2023 - 2003 = 20 years
- 2006: 2023 - 2006 = 17 years
- 2007: 2023 - 2007 = 16 years
- 2011: 2023 - 2011 = 12 years

Arranging the ages in descending order (from eldest to youngest):
42, 40, 36, 29, 26, 24, 23, 20, 17, 16, 12

Now, let's apply the given conditions step-by-step:

1. Age of L: The age of L is a prime number. The prime numbers in the list of ages are 29, 23, and 17.
2. L and Q: The age difference between L and Q is 3 years. This gives the following possibilities:
- If L = 29, Q = 26
- If L = 23, Q = 20 or 26
- If L = 17, Q = 20

3. P and Q: P is three persons younger than Q (i.e., P is at 3 positions below Q in the descending age list).
- If Q = 26 (5th position), P must be at the 8th position, which is 20.
- If Q = 20 (8th position), P must be at the 11th position, which is 12.

Let's check the case where Q = 20, P = 12, and L = 23:
- O and L: O is either 5 persons elder or 5 persons younger than L. L (23) is at the 7th position. Five positions elder is the 2nd position, which is 40. So, O = 40.
- O and N: The number of persons elder to O is one less than the number of persons younger to N. There is 1 person elder to O (40). Thus, the number of persons younger to N must be 2. This means N is at the 9th position, which is 17.
- T and N: T is immediately younger to N. So, T is at the 10th position, which is 16.
- R and T: The age difference between R and T is 13 years. Since T = 16, R = 16 + 13 = 29.
- R and Q: R (29) is not immediately elder to Q (20), which satisfies the condition.
- S and K: S is 2 years elder to K. The remaining available ages are 42, 36, 26, and 24. The only pair with a difference of 2 is 26 and 24. Thus, S = 26 and K = 24.
- M: The age of M is completely divisible by 4 and M is elder to K (24). The remaining ages are 42 and 36. Since 36 is divisible by 4, M = 36.
- U: The only remaining age is 42, so U = 42.

Let's calculate the product of the ages of U and Q:
$\mathrm{Product}=42\times 20=840$

The correct answer is 840.