Nazari, the violinist's age

Question

"How old are your five daughters?", my neighbour Ignacio asked me this morning.

"They are all under 20, and their ages add up to a prime number", I answered.

"Tell me more", said Ignacio.

"Well, if you were to choose any two of them, no matter which two, their ages would have a common divisor greater than 1", I told Ignacio.

"I still can't be certain how old they are", complained Ignacio.

"And you still couldn't know even if I told you how old they are altogether, nor indeed, if I also told you, besides, that the eldest, Nazari, plays the violin beautifully", I responded.

"Fine, but now at least I know for sure how old Nazari is!", Ignacio joyfully said after a long pause.

How old is Nazari, the eldest of my daughters?

Answer 1

There's been a change to the puzzle. The new meaning is that the sum of the ages is insufficient, but knowing the sum and that the eldest has a unique age is sufficient.

Possible age sets:

 43   6  6  6 10 15
 47   6  6 10 10 15
 53   6 10 10 12 15
 59   6 10 10 15 18
 59  10 10 12 12 15
 61   6 10 12 15 18
 61   6 10 15 15 15
 61  10 12 12 12 15
 67   6 10 15 18 18
 67  10 12 12 15 18
 67  10 12 15 15 15
 71  10 10 15 18 18
 73  10 12 15 18 18
 73  10 15 15 15 18
 79  10 15 18 18 18

Removing the lines where the sum of ages is sufficient, and putting an X where the eldest has a nonunique age:

 59   6 10 10 15 18
 59  10 10 12 12 15

 61   6 10 12 15 18
 61   6 10 15 15 15 X
 61  10 12 12 12 15

 67   6 10 15 18 18 X
 67  10 12 12 15 18
 67  10 12 15 15 15 X

 73  10 12 15 18 18 X
 73  10 15 15 15 18

If the sum is 59 or 61, even knowing there's a unique eldest isn't enough. But it might be if the sum is 67 or 73.

 67  10 12 12 15 18
 73  10 15 15 15 18

So Nazari is 18, and two other girls are 10 and 15. But I can't see a way to tell whether the remaining two are both 12 or both 15.

Answer 2

15 or 18 both seem possible. The intended answer seems to be 15

None of their ages can be a prime number or power of prime, since then every other daughter must have a multiple of that prime, and so the sum will be divisible by it.

Thus each age must the product of two different primes. The 20 year limit means that these primes must be 2, 3, and 5. At least one of these factors must be absent from one of the ages, so a 3 x 5 = 15 age must exist (and so 7 would be too big). The set of possible ages are 6, 10, 12, 15, and 18, meaning the eldest might be 15 or 18.

From here, we must use the knowledge that the sum uniquely determines who is the eldest. I don't see any particularly clever way of doing this, but a computer search shows that there are 5 valid combinations of ages, as listed in Laska's answer. Those for which the eldest is 18 are (10, 10, 15, 18, 18) and (10, 15, 18, 18, 18). For the others, the eldest is 15.

From there, we have to assume the poster lives somewhere where adoption is illegal, everyone is monogamous, pregnancies can't occur within 4 months of a birth, and twins, triplets, etc are never dissociated by birth order. In that case, the 18 situation is ruled out.

Answer 3

Nazari is 15.

No daughter can be 1, because then there is no common factor. No daughter can be p^a, a prime power, because then all daughters must have p as a factor, and p will divide the sum S.

No daughter has age divided by 3 primes, since that would mean her age is at least $2 \times 3 \times 5 = 30$. So every daughter has age divided by 2 primes. There must be at least 3 primes dividing the product of the ages, since otherwise each daughter's age is divisible by the same two primes, and S is divisible by this composite.

Each of these 3 primes p,q,r is missing from at least one daughter's age, since otherwise S shares a prime factor. So at least one daughter has age divisible by pq another by pr and a third by qr.

Wlog assume p<q<r. Then since each age is less than 20 we know that p=2, q=3 & r=5. And one daughter is exactly 10 and another is 15.

So far, there is just a single path of logic. However there seem to be a bunch of possibilities for the ages of the other 3 daughters, subject to the constraint that one has an age divisible by 6. The available ages are 6,10,12,15,18.

Since the sum of the daughters' ages is prime, it is odd. So 0 or 2 of these three daughters have odd ages. The only odd age is 15, so we can have 12,15,15,15 and the last one is 6, 12 or 18. Three possibilities

Alternatively, we have none of the three being 15, and there are 19 ways that 3 daughters can have ages drawn from {6,10,12,18}, if we exclude the possibility that all are aged 10.

So 3+19=22 possibilities, from which we exclude the other possibilities where S is composite odd, and where S is non-unique (since knowing S will give us all the ages).

There are then 5 possibilities remaining:

We are meant to assume that Nazari is neither a twin nor a triplet, so she cannot be 18. There remain three other possibilities, and in all of them Nazari is 15. We cannot deduce the ages of all her sisters, but we know that one is 6 and one is 10. The others are 6 & 6, 6 & 10 or 10 & 12.