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Continued Multiples

- 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.
- 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.

Solution

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