## Task 2 - 300722

a) Determine among the natural numbers $a$ that are greater than 100 and less than 1000, all those that satisfy the following conditions (1), (2), (3)!

(1) $a$ has exactly two different natural numbers as divisors.

(2) $a$ leaves a remainder of 2 when divided by 11 and also when divided by 13.

(3) $a$ is an odd number.

b) For each of the three conditions (1), (2), (3), determine whether the result of part (a) changes if this condition is omitted and only the other two conditions are required!
a) For every natural number $a$ that satisfies the first sub-statement in (2), $a-2$ is a multiple of 11, i.e., one of the numbers $0, 11, 2 \cdot 11, 3 \cdot 11, \ldots$

If $a$ also satisfies the second sub-statement in (2), then $a-2$ is also a multiple of 13, so only the numbers $0, 13 \cdot 11, 2 \cdot 13 \cdot 11, 3 \cdot 13 \cdot 11, \ldots$ remain for $a-2$.

If $a$ also satisfies (3), then $a-2$ is an odd number, so only the numbers 143, $3 \cdot 143 = 429, 5 \cdot 143 = 715, 7 \cdot 143 = 1001, \ldots$ remain for $a-2$.

Among the natural numbers $a$ with $100 < a < 1000$, only the numbers $143 + 2 = 145, 429 + 2 = 431$, and $715 + 2 = 717$ satisfy the conditions (2) and (3).

Since $145 = 5 \cdot 29$, 145 has more than two distinct natural numbers as divisors, namely 1, 5, 29, 145. The same applies to 717 because $717 = 3 \cdot 239$. On the other hand, 431 is a prime number. Therefore, 431 has exactly the two natural numbers 1 and 431 as divisors.

Thus, among the natural numbers $a$ with $100 < a < 1000$, the conditions (1), (2), and (3) are satisfied exactly by the number $a = 431$.

b) As shown above, (2) and (3) are also satisfied, for example, by 145. Furthermore, (1) and (3) are, for example, also satisfied by the odd prime number 101.

If one of the conditions (1) or (2) is omitted, the result from a) changes.

However, (3) can be omitted without changing the result; because every natural number that satisfies (1) is a prime number, and if it is greater than 100, the only even prime number 2 is excluded, i.e.: For numbers $a$ that are greater than 100 (and less than 1000), (3) already follows from (1).