## Task 1 - 140711

Klaus claims that he has exactly 17 coins in his wallet with a total value of 34 pfennigs.

Determine all possibilities for the quantities of each type of coin that Klaus could have in this case! It is assumed that only coins of the current currency of the GDR are to be considered.
The specified amount can only consist of 1-pfennig, 5-pfennig, 10-pfennig, and 20-pfennig coins, as the current currency of the GDR does not have any other coins up to a value of 34 pfennigs.

Assuming Klaus had a 20-pfennig coin, the remaining 16 coins would have to total 14 pfennigs. This is not possible. Therefore, Klaus cannot have a 20-pfennig coin in his wallet.

Assuming Klaus had more than one 10-pfennig coin, he could have at most 15 more coins, whose total value could be at most 14 pfennigs; this is again not possible. Thus, Klaus can have at most one 10-pfennig coin in his wallet.

Assuming Klaus had no 10-pfennig coins. If he had 5-pfennig coins in the amount of $x$, then he would have 1-pfennig coins in the amount of $17-x$, and it would be $5x + 17 - x = 34$, thus $4x = 17$. This is not possible, as 17 is not divisible by 4.

Therefore, the only remaining possibility is that Klaus has exactly one 10-pfennig coin and the rest of the amount in other coins. If he has 5-pfennig coins in the amount of $x$, then he has 1-pfennig coins in the amount of $16 - x$. This results in $5x + 16 - x = 24$, thus $4x = 8$ and therefore $x = 2$. If Klaus's claim is correct, he must have exactly one 10-pfennig coin, exactly two 5-pfennig coins, and exactly fourteen 1-pfennig coins in his wallet.