## Task 1 - 220831

On a day in 1981, Cathrin asks her grandfather about his birth year. The grandfather, a friend of puzzle questions, replied:

"I am older than 65 years, but younger than 100 years. The year of my birth is not divisible by 2, 3, or 5. The remainder when this year is divided by 60 is not a prime number."

Investigate whether these statements can all be true for a birth year and whether they uniquely determine the birth year! What is the birth year of the grandfather?

Hint: The year should be given in full, e.g., not 11 but 1911.
I. If the information for a birth year applies, then:

Since the grandfather was older than 65 years and younger than 100 years on a day in 1981, he was born before the corresponding date in 1916 and after the corresponding date in 1881.

The year of his birth is therefore one of the numbers 1881, 1882, ..., 1916.

Of these, only the following are not divisible by 2, 3, or 5:

$$
1883, 1889, 1891, 1897, 1901, 1903, 1907, 1909, 1913
$$

These numbers leave the following remainders when divided by 60: $23, 29, 31, 37, 41, 43, 47, 49, 53$.

Of these, only 49 is not a prime number. Therefore, the information can only apply to the birth year 1909.

II. This applies here; for if the grandfather was born in 1909, he was either 71 or 72 years old on a day in 1981, thus older than 65 and younger than 100 years; furthermore, 1909 is not divisible by 2, 3, or 5, and leaves a remainder of 49 when divided by 60, which is not a prime number.

From I. and II. it follows: The information can apply in total, and it uniquely determines the birth year. It is 1909.