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15 or 18 both seem possible

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. The largest possible valid sum is 10, 15, 15, 15, 18, yielding 73. It is clearly impossible to achieve this without the 18. But alas, a unique sum can also be found with 15 as the eldest. 6, 6, 10, 10, 15 yields a sum of 47. I'll check with a computer when the opportunity arises, but this also seems unique.

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.

15 or 18 both seem possible

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 eldest might also be 18 (3 × 3 × 2).

An example to demonstrate existence is 6, 6, 10, 10, 15 for a sum of 47. To include an 18, we can have 6, 10, 10, 15, 18 for a sum of 59.

It is unclear to me why the statement about Nazari matters.

15 or 18 both seem possible

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. The largest possible valid sum is 10, 15, 15, 15, 18, yielding 73. It is clearly impossible to achieve this without the 18. But alas, a unique sum can also be found with 15 as the eldest. 6, 6, 10, 10, 15 yields a sum of 47. I'll check with a computer when the opportunity arises, but this also seems unique.

15 or 18 both seem possible

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 eldest might also be 18 (3 × 3 × 2).

An example to demonstrate existence is 6, 6, 10, 10, 15 for a sum of 47. To include an 18, we can have 6, 10, 10, 15, 18 for a sum of 59.

It is unclear to me why the statement about Nazari matters, the eldest would have to be 15 regardless.

15 or 18 both seem possible

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 eldest might also be 18 (3 × 3 × 2).

An example to demonstrate existence is 6, 6, 10, 10, 15 for a sum of 47. To include an 18, we can have 6, 10, 10, 15, 18 for a sum of 59.

It is unclear to me why the statement about Nazari matters.