Question

Suppose Haruka has a special key Δ in her calculator called delta key:
Rule 1: If the display shows a one-digit number, pressing delta key
Δ replaces the displayed number with twice its value.
Rule 2: If the display shows a two-digit number, pressing delta key
Δ replaces the displayed number with the sum of the two digits.
Suppose Haruka enters the value 1 and then presses delta key
Δ repeatedly. After pressing the Δ key for 68 times, what will be the displayed number?

• A. 7
• B. 4
• C. 10
• D. 2
• E. 8

Answer 1

The Correct Option is A

To solve the problem of determining the number displayed after pressing the delta key (Δ) 68 times, follow these steps:

1. Initially, Haruka enters the number 1. This is a one-digit number.
2. Applying Rule 1: If the display shows a one-digit number, pressing delta key Δ replaces the displayed number with twice its value.
3. First Press: The number changes from 1 to 2 (twice of 1).
4. Second Press: The number changes from 2 to 4 (twice of 2).
5. Third Press: The number changes from 4 to 8 (twice of 4).
6. Fourth Press: Since 8 is a one-digit number, apply Rule 1 again, changing the number from 8 to 16 (twice of 8).
7. Fifth Press: The number is now 16, a two-digit number. Apply Rule 2: If the display shows a two-digit number, pressing delta key Δ replaces it with the sum of the two digits.
8. Sixth Press: The sum of digits in 16 is 1 + 6 = 7.
9. Seventh Press: The number is now 7, a one-digit number. Applying Rule 1, it changes to 14 (twice of 7).
10. Eighth Press: Apply Rule 2 on 14, and the sum of digits is 1 + 4 = 5.
11. Ninth Press: The number is now 5; applying Rule 1 changes it to 10 (twice of 5).
12. Tenth Press: Apply Rule 2 on 10, and the sum of digits is 1 + 0 = 1.
13. This cycle of transformations continues:
14. Cycle Repeats: After obtaining 1 again, the next values are 2, 4, 8, 16, 7, 14, 5, 10, and then back to 1. This cycle has a length of 9 transformations.
15. Calculate Total Cycles: With 68 presses, determine how many complete cycles and extra presses occurred. Dividing 68 by 9 gives a quotient of 7 (representing full cycles) with a remainder of 5 (extra presses into the next cycle).
16. Performing 5 extra presses beginning from 1 results in the sequence: 1 → 2 → 4 → 8 → 16 → 7.

Thus, after pressing the delta key 68 times, the displayed number is 7.