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6*7*7*7...........

Discussion in 'General Discussion' started by IsaiahX, Nov 17, 2018.

  1. IsaiahX

    IsaiahX Dreamer and Oneironaut

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    Have you ever noticed that if you multiply 6 by 7, and keep multiplying it by 7, and then add all the digits in that number together to one digit, it's 6 every time?


    Let me add a few examples:


    6 times 7 is 42, 4 plus 2 is 6


    42 times 7 is 294, 4 plus 9 is 13, 1 plus 3 is 4, and 4 plus the original 2 in the hundreds place is 6


    294 times 7 is 2058, 8 plus 5 is 13, 1 plus 3 is 4, plus 2 is 6


    2058 times 7 is 14406. 6 plus 4 is 10, 1 plus 0 is 1, plus 4 is 5, plus 1 is 6


    14406 times 7 is 100842. 2 plus 4 is 6 plus 8 is 14, 1 plus 4 is 5, plus 1 is 6


    100842 times 7 is 705894. 4 plus 9 is 13. 1 plus 3 is 4. 4 plus 8 is 12. 1 plus 2 is 3. 3 plus 5 is 8. 8 plus 7 is 15. 1 plus 5 is 6


    Just an interesting thing I wanted to share.
     
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  2. Brickjectivity

    Brickjectivity Veteran Member
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    There is also a similar principle with the number nine. If nine is a factor then you can use it to check certain math operations. This is called 'Casting out nines'. 6 times any number means six of those numbers.

    Eleven is the number with the greatest number of strange properties. It has causes some mathematicians to propose a new number system (base 12) called duodecimal. Apparently if we all learned duodecimal instead of base 10 we would be able to do arithmetic easier.
     
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  3. Polymath257

    Polymath257 Think & Care
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    Much more than this is true. The point here is that 7 leaves a remainder of 1 when divided by 6.

    Start with 6 and multiply by *any* sequence of numbers which give a remainder of 1 when divided by 6. Then do your adding procedure. You will always get back to 6.

    So, for example, 25, 55, and 121 all leave remainders of 1 when divided by 6. Take 6 and multiply by all of these: 6*25*55*121=998250. Add up the digits: 33. Add up again: 6.

    /E: the same is true if you replace 6 by 3 everywhere.
     
    #3 Polymath257, Nov 17, 2018
    Last edited: Nov 17, 2018
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  4. 74x12

    74x12 Well-Known Member

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    The number 9 thing is interesting but largely based on the fact we go by a 9 numeral system. If you were to change to a 6 numeral system then 6 would be the number with this strange property.

    0
    1 2 3 4 5 6
    10
    11 12 13 14 15 16
    20
    21 22 23 24 25 26
    30
    etc ...
     
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  5. Brickjectivity

    Brickjectivity Veteran Member
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    So it would work with any base or just 6 and 9?
     
  6. 74x12

    74x12 Well-Known Member

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    Actually I spoke too soon. Eventually if you keep going with 6 you end up with 12 which of course becomes 3. So I guess 9 is a special number after all. Still interesting how it plays out.
     
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  7. Wirey

    Wirey Fartist

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    This is a base of three thing. All numbers divisible by six are divisible by three. All even numbers divisible by three are divisible by six, as by their structure they must be divisible by three and two. If you look at three as a constant, you can see that every time you raise the integer of the ten (i.e. going from 9 to 12) the new two digit number, if added as separate digits, will be a multiple of three.
     
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  8. Polymath257

    Polymath257 Think & Care
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    What is going on is *modular* arithmetic base 9. So, think of a clock with only 9 hours instead of 12. We can add or multiply by doing regular multiplication, but then taking the remainder after dividing by 9.

    If we use base 10 arithmetic, we get the same result modulo 9 after we add up the decimal digits. This is because all powers of 10 have a remainder of 1 when divided by 9. Now, all numbers that have a remainder of 1 when divided by 3 or 6 will be 1,4, or 7, modulo 9. Any of those multiplied by 3 are 3 modulo 9. The same is true for 6.

    This is the underlying reason for what you found.
     
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