Continued Multiples

  1. Take a 2-digit number and multiply the two digits together. Repeat with each product until a single digit is the result. For most starting numbers, no more than three of these steps are needed to arrive at the final single digit. (For example: 57: 5*7=35: 3*5=15: 1*5=5). Find one that takes four steps.
  2. Start with a 3-digit number. Choose any one of the digits and remove it to make a 2-digit number. Multiply the 2-digit number by the digit you removed. Eventually, you will get a 2-digit number, which will yield a single digit, as above. What number should you start with, and what steps should you take to find the longest series of steps you can take to arrive at that final single digit?
Feel free to generalize or expand on the above.

Source: Previous puzzle experience.

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