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Take any two digit number none of whose digit is $0$. Now add the product of the digits in that number. if number becomes three digit number take last two digits. you will find a chain of numbers or a multiple of 10 after repeating this process: $02$, $04$, $08$, $16$, $22$, $26$, $38$, $62$, $74$, $102$, $04$......... Can anyone help me to prove this?

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Hint: The numbers in the chain are integers from 0 to 99. What will happen when you compute 101 numbers (or maybe before)?

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