# 7.5. (7 points)

In a family, there are six children. Five of them are older than the youngest by 2, 6, 8, 12, and 14 years, respectively. How old is the youngest if the ages of all the children are prime numbers?
Answer: 5 years.

Solution. First, let's check the prime numbers less than six. It is obvious that the number we are looking for is odd. The number 3 does not satisfy the condition because $3+6=9-$ is not a prime number. The number 5 satisfies the condition because $5+2=7, 5+6=11, 5+8=13, 5+12=17, 5+14=19$, which means the ages of all the children are indeed prime numbers.

Let's prove that this answer is unique. For this, consider the remainders when the numbers added to the age of the youngest child are divided by 5: the remainder of 2 is 2, the remainder of 6 is 1, the remainder of 8 is 3, the remainder of 12 is 2, and the remainder of 14 is 4. Thus, all remainders from 1 to 4 are present. Therefore, if the age of the youngest child is not a multiple of five, adding a number with a remainder that complements it to five will result in a number that is a multiple of five and greater than five, which is a composite number.