## Task 2 - 260722

Klaus met Dorit at a mathematics specialist camp and asked her about her age. She replied:

"I turned 10 years old in May of the year whose number is the smallest number divisible by 7 that leaves a remainder of 1 when divided by 2, 3, 5, and 11."

Determine whether Klaus could uniquely determine Dorit's age from this answer. If this is the case, then state how old (in full years) Dorit was in June 1986!
Dorit turned ten years old in the year $x$. Then $x-1$ is a natural number that is divisible by $2, 3, 5$, and 11. These four divisibility statements are true if and only if $x-1$ is a multiple of the least common multiple (LCM) of these four numbers. This is equivalent to saying that with a natural number $n$

$$
x-1=330 \cdot n \quad \text { i.e. } \quad x=330 \cdot n+1
$$

Since 330 leaves a remainder of 1 when divided by 7, $330 \cdot 2$ leaves a remainder of 2, $330 \cdot 3$ leaves a remainder of 3, and so on, $n=6$ leads to the smallest number $x$ that (in addition to the mentioned divisibility conditions for $x-1$) also satisfies the condition of being divisible by 7.

This results in $x=330 \cdot 6+1=1981$; hence, from Dorit's answer, it can be uniquely determined:

Dorit turned ten years old in May 1981; by June 1986, she is thus 15 years old.