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Math Help - f(a)=100, f(124a)=700 -> f(3a)=?

  1. #1
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    f(a)=100, f(124a)=700 -> f(3a)=?

    Let f(n) denote the sum of the digits of the positive integer n in decimal notation. What may f(3a) be if f(a)=100 and f(124a)=700?
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  2. #2
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    Hello, james_bond!

    Is this a trick question?


    Let f(n) denote the sum of the digits of the positive integer n in decimal notation.
    What may f(3a) be if f(a)=100 and f(124a)=700?

    Anyone familiar with "Casting out 9's" know this fact . . .
    . . If f(n) is the digital-sum function: . f(a\cdot b) \:=\:f(a)\cdot f(b)

    (Hmm, that's not quite accurate, but it'll do.)


    If f(a) = 100, then: . f(3a) \;=\;f(3)\!\cdot\!f(a) \;=\;3\cdot100 \;=\;300 . . . . too easy

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  3. #3
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    It wasn't supposed to be a trick question and I think that you're wrong since for example f(9)f(9)=81\ne f(9*9)=f(81)=9.
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  4. #4
    Senior Member JaneBennet's Avatar
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    I know. a is the 100-digit number consisting of all 1ís, so 124a is the 102-digit number whose first two digits are 13 and last two 64, with 98 7ís in between. Hence f(3a) = 300 after all.
    Last edited by JaneBennet; December 22nd 2007 at 06:26 AM. Reason: Minor formatting typo
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  5. #5
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    Quote Originally Posted by JaneBennet View Post
    I know. a is the 100-digit number consisting of all 1ís, so 124a is the 102-digit number whose first two digits are 13 and last two 64, with 98 7ís in between. Hence f(3a) = 300 after all.
    Very neat! But can you be sure that this is the unique solution???
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  6. #6
    Senior Member JaneBennet's Avatar
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    Well, I’ve answered the question (what may f(3a) be) so I’m happy with that. There may be other answers, I don’t know, but I don’t think they are as neat as this simple one.
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