Mathematical parents 2

Note that Zachary is 9 years younger than Yasmine, 13 years younger than Xavier and 14 years younger than Wendy. So two possible scenarios:

• Wendy (odd), Xavier (even), Yasmine (even), Zachary (odd)
• Wendy (even), Xavier (odd), Yasmine (odd), Zachary (even)

In general, we can ignore the even ages considering that the only even prime is 2.

Employing the hint, an odd number that is less than 9 is a prime. This is because if $n<9$ , then $\sqrt{n} <3$ where $3$ is the next prime after $2$ . Thus the smallest odd composite number is $9$ . Thus in the former case, the smallest possible age for Zachary would be 9. That means Yasmine would be 18, Xavier 22 and Wendy 23. However, 23 is a prime number. The next smallest possible age for Zachary would be 15 but then Wendy would be 29 which is also a prime number.

Similarly, an odd number not divisible by 3 that is less than 25 is a prime. In this case, $n<25$ so that $\sqrt{n}<5$ where $5$ is the next prime after $2$ and $3$ . Considering the latter case now, if both Xavier and Yasmine are under 25, at least one of them has an age that is a prime number. This is because their ages differ by 4 so both ages are odd and at least one of them is not divisible by 3, hence a prime. On the other hand, if Xavier is 25, then Yasmine would be 21 which is sufficient to satisfy the given condition considering that 25 and 21 are composite numbers. Clearly, Wendy and Zachary's ages are even integers greater than 2, hence composite integers too. Respectively, they are 26 and 12.

Thus our answer is $26+25+21+12=\boxed{84}$ . This is indeed the very first time all ages are composite numbers.