Directions: Study the following information carefully and answer the questions…

Concept

In a coded age-and-relations puzzle, the asked value is fixed by intersecting (a) a generation/role deduction from the family clues with (b) the arithmetic clues on age differences. A clue of the form "X's age is prime and X is not the oldest" narrows the candidate ages; the remaining ambiguity is then broken by a second, independent arithmetic clue, not by which numbers happen to appear as choices.

Application

First convert birth years to ages on the base year 2023 (age = 2023 - birth year):

Now fix the family structure and the three generations:

1. M has a son, no sibling, and a father; M's father has no sibling. P is the paternal grandmother of M's only son, which makes P the mother of M — taking M as male, since the puzzle's single grandmother-figure role (P) only works cleanly this way, without needing a ninth, unmentioned relative for a separate "M's own mother." So the eldest generation is P, M's father, and one more elder; the three eldest ages are 71, 67 and 61.

2. D is the one whose own numeric clue supplies the missing parent-child age gap: D has no sibling, matching M's father's no-sibling trait, and the clue "M's uncle is not D" only needs stating if D could otherwise be mistaken for that eldest-generation seat — so D is M's father, and D's son is M (M has no sibling of his own, so M is D's only child). This means D's clue — the age gap between D and D's son is prime and under 30 — applies directly to the M's father-M pair.

3. Pin down M's own age. The middle generation holds 42, 39 and 36, shared by M, M's spouse and N's mother. The multiple-of-10 clue between N and N's mother allows N's mother = 39 (N = 9) or N's mother = 36 (N = 6), leaving {42, 36} or {42, 39} respectively for M and M's spouse. Test each non-42 candidate against the D-clue: M = 36 gives gaps 71-36=35, 67-36=31, 61-36=25 to the three eldest ages — none is both prime and under 30 (31 is prime but not under 30; 35 and 25 are not prime); M = 39 gives 71-39=32, 67-39=28, 61-39=22 — none is prime. So neither 36 nor 39 can be M; M is 42 (71-42=29 and 61-42=19 are both prime and under 30).

4. Identify B as M's spouse using the sibling and spousal-gap clues: with the spouse candidate at 36, B's only-sibling gap is |36-39| = 3, matching (1/3) x age(N) = 9/3 = 3 when the sibling is S = N's mother (39) and N = 9; and the spousal gap |36-42| = 6 matches 2 x Diff(M, S) = 2 x |42-39| = 6. Both check out, so B is M's spouse (36), and S is B's own sibling — consistent with "S is not married to B."

5. Since M's father has no sibling, M's uncle must be a sibling of M's mother P. M's uncle therefore belongs to the eldest generation, and the clue states his age is prime and he is not the oldest, so his age is 61 or 67 (the oldest, 71, is excluded).

6. As established, D is M's father, so the D-clue's prime-and-under-30 age gap applies to M's father and M directly. With M = 42, the only eldest-generation age giving such a gap is 71 (29) or 61 (19); an age of 67 gives 67 - 42 = 25, which is not prime. Hence M's father is 71 or 61, never 67.

7. Confirm the spouse and break the 61-vs-67 tie with the mother-in-law clue: the gap between B (M's spouse) and her mother-in-law (M's mother, P) must be a multiple of 5. If B were 39: |39-71|=32, |39-67|=28, |39-61|=22 — none is a multiple of 5, so B cannot be 39 (confirming B is 36, as verified above, with N's mother at 39 and N at 9). With B at 36: |36-67|=31 is not a multiple of 5, while |36-61|=25 and |36-71|=35 are. So P cannot be 67; P is 61 or 71.

8. Combine: 67 is not M's father (step 6) and 67 is not P (step 7). The only eldest-generation seat left for 67 is M's uncle, and 67 is prime and not the oldest, satisfying every condition. So M's uncle is 67.

Cross-check

• 67 is prime and is not the maximum age (71 is), matching "prime but not the oldest".

• Difference clue stays consistent: M's spouse 36 with spouse M 42 gives a gap of 6, equal to twice the M-S gap (S = 39, so 2 x |42 - 39| = 6).

• M's father then takes 71 (71 - 42 = 29, prime) and P takes 61 (|36 - 61| = 25, a multiple of 5); every clue holds with M's uncle at 67.

Answer: 67 years.