Mathematical parents 1

Number Theory Level pending

A couple has four children - Wendy, Xavier, Yasmine and Zachary. What's interesting about these children is as follows:

Wendy is $1$ year older than one of her younger brothers, Xavier.
Xavier is $2$ years older than his younger sister, Yasmine.
Yasmine is $3$ years older than her younger brother, Zachary.

For the first time when the ages of all four children are composite numbers, what is the combined sum of their ages?

Hint : If an integer $n$ is not divisible by any prime $p \leq \sqrt{n}$ , then $n$ is a prime integer.

If you like this problem, try another one here .

The answer is 50.

1 solution

Noel Lo
Jul 19, 2018

Note that Zachary is 3 years younger than Yaasmine, 5 years younger than Xavier and 6 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. Then Yasmine would be 12, Xavier 14 and Wendy 15, all composite numbers.

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 2 so both ages are odd and at least one of them is not divisible by 3, hence a prime.

Thus our answer is $15+14+12+9=\boxed{50}$ . This is indeed the very first time all ages are composite numbers.

Mathematical parents 1

The answer is 50.

1 solution

0 pending reports