Joseane's calculator has gone crazy: for each digit she presses, its double appears on the screen. The addition, subtraction, multiplication, and division operation keys work normally and cannot be pressed twice in a row. For example, a permitted sequence of operations is to write $2 \rightarrow \times \rightarrow 3$, which generates the number $4 \cdot 6=24$.

a) How can she make 80 appear by pressing 3 keys?

b) How can she make 50 appear by pressing 3 digit keys and two operation keys alternately?

c) What is the minimum number of keys she must press to obtain the number 23?
Solũ̧̧̃o

a) By pressing the digits, the numbers that can appear on the screen are $0, 2, 4, 6, 8$, 10, 12, 14, 16, and 18. One way to make 80 appear on the display is to press the sequence $4 \rightarrow \times \rightarrow 5$, which results in $8 \times 10=80$.

b) The sequence should be: digit - operation - digit - operation - digit. One way is $5 \rightarrow \times \rightarrow 5 \rightarrow \div \rightarrow 1$, which results in the operation $(10 \times 10) \div 2=50$.

c) One way to get 23 is to press 4 keys: $2 \rightarrow 3 \rightarrow \div \rightarrow 1$, which results in $46 \div 2 = 23$. Since all the digits entered generate even numbers, if we use only the keys +, -, or $\times$, the result will be even, and thus she will not obtain 23. Therefore, the symbol $\div$ will be used at least once. If she uses another operation key, as they cannot be pressed twice in a row, at least 5 keys will need to be pressed. On the other hand, using only one operation, we will need at least two more keys with digits. Since no admissible division between the digits in the set $\{0, 2, \ldots, 18\}$ produces 23, the minimum number of keys she must use is 4.