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Math Help - palindromic numbers

  1. #1
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    palindromic numbers

    A palindromic number is one that reads the same from left to right or from right to left. For example, 64746 is a palindromic number.

    a/ How many odd six-digit palindromic numbers are there?
    b/ How many odd seven digit palindromic numbers are there in which every digit appears at most twice?
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  2. #2
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    How many odd 1-, 2-, and 3-digit numbers are there?
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  3. #3
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    Wink Thanks

    Thanks for your time, but I worked out the answer myself
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  4. #4
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    Well now you've got me curious. Do you mind showing us how you solved it?

    Thanks
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  5. #5
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    Quote Originally Posted by mathsmad
    A palindromic number is one that reads the same from left to right or from right to left. For example, 64746 is a palindromic number.

    a/ How many odd six-digit palindromic numbers are there?
    b/ How many odd seven digit palindromic numbers are there in which every digit appears at most twice?
    i didn't understand if in both a and b there is each digit only twice so i will solve twice

    a/ <each digit twice>
    the sum of chosing including leading zero minus the sum of chosing with leading zero
    5*9*8-9*8=288

    <as many digits>
    10*9*8-9*8=648

    b/
    5*9*8*7-9*8*7=2016
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  6. #6
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    Consider the slots:
    _ _ _ _ _ _
    In the first slot you can only place 5 numbers (because since it is odd the last slot must be 1,3,5,7, or 9).
    In the second slot you can place 10 numbers.
    In the third slot you can place 10 numbers.
    In the fourth slot you can only have one number (same as third slot because it is palindromic).
    In the fifth slot you can only have one number.
    In the sixth slot you can only have one number.
    Thus by the Fundamental Counting Principle there are a total of 5*10*10 possibilities thus, 500 different numbers.

    One interesting fact about palindromes is that if it is even digited then it is always divisible by 11. And if it is odd digited then is remainder when divided by 11 is the middle digit (thus if it is zero then it is divisible by 11 and this type of number can never be have remainder 10). Thus "there does not exist a palindromic number when divided by 11 leaves a remainder of 10!"
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by mathsmad
    A palindromic number is one that reads the same from left to right or from right to left. For example, 64746 is a palindromic number.

    a/ How many odd six-digit palindromic numbers are there?
    b/ How many odd seven digit palindromic numbers are there in which every digit appears at most twice?
    Consider an odd number <=999. Each of these gives a unique six-digit
    palindromic number as follows:

    first add as many zeros to the front of the number as needed so that we
    have exactly three digits. Reverse the digits and append reversed copy to
    the front of the three digits.

    It is self evident that any six-digit palindromic number can be generated in
    this manner.

    Therefore there are exactly as many six-digit palindromic numbers as there
    are odd numbers <=999. There are exactly 500 odd numbers <=999, so
    there are exactly 500 six-digit palindromic numbers.

    For seven-digit odd palindromic numbers the three most and the three least
    significant digits are the digits of a six-digit palindromic number. Thus for
    each possible middle digit there are 500 seven-digit palindromic numbers. The
    possible middle digits are 0, 1, 2, .., 9. Hence there are 5000 seven-digit
    odd palindromic numbers.

    RonL
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  8. #8
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    Quote Originally Posted by mathsmad
    A palindromic number is one that reads the same from left to right or from right to left. For example, 64746 is a palindromic number.
    Hello,

    try this one about palindromic numbers:

    You start with any number, which isn't a palindromic one. You write the ciphers of this number from end to start and add it to the original number. If the result is palindromic than you've got what you wanted. If the result is not a palindromic number you write the ciphers (of the result of course!) from end to start, add ... proof ... and so on, until you get a palindrom.

    I've attached a simple example. If you start for example with 40793 than you need 22 steps to get a palindrom.

    I don't know if this rule is proofed (I never found a proof for it). So maybe the proof is a nice job for a boring weekend ;-)

    Bye
    Attached Thumbnails Attached Thumbnails palindromic numbers-drehzahl2.gif  
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