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Math Help - Osculator

  1. #1
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    Question Osculator

    Hello Everyone

    Can anyone explain why we cannot use 'osculators' to check divisibility for an even number or a number ending with 5?
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  2. #2
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    What is an "osculator"?
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  3. #3
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    Ok, I found the answer in another site. Just copy-pasting the info here for informational purpose.

    Osculator Method/Seed of a number

    Osculator/seed of a number can be negative and positive
    Let the number be n,then if nk+1=10m then positive osculator of n is m
    And if nk-1=10m then the negative osculator of n is m
    Thus if the number is 7
    We have 7(3)=21 and 21-1=10(2) so the negative osculator of 7 is 2
    We have 7(7)=49 and 49+1=10(5) so the positive osculator of 7 is 5

    This osculator is used to check for the divisibility by 7 , 13 and other prime numbers for which there are no straight forward rules.

    Osculator for an even number or a number ending with 5

    If the number is even then nk+1 and nk-1 are odd which cant be divisible by 10,hence the osculator of an even number doesnít exist

    Similarly when a number ends with 5,then also the osculator is not possible

    Thus only odd numbers have their osculators and which arenít also multiples of 5.

    Examples :
    1) Lets see whether 7896834 is divisible by 7 or not?
    The negative osculator of 7 is 2
    We will do the following operations on the number
    789683 -4(2)=789675
    78967-5(2)=78957
    7895-7(2)=7881
    788-1(2)=786
    78-6(2)=66
    6-6(2)=-6 which is not divisible by 7
    Hence the number is NOT divisible by 7

    Infact the thing which we are doing is this
    7896834mod7=>(7.10^6 + 8.10^5 + 9.10^4 + 6.10^3 + 8.10^2 + 3.10^1 + 4.10^0 mod7)
    Now gcd(7,10)=1
    So we can find integers r and s such that 7r+10s=1
    We know 7(3)-1=10(2)=> the negative osculator of 7 is 2
    7(3)+10(-2)=1 hence the value of s=-2

    Then we can say that
    (7.10^6 + 8.10^5 + 9.10^4 + 6.10^3 + 8.10^2 + 3.10^1 + 4.10^0 mod7)
    =>(7.(-2)^6 + 8.(-2)^5 + 9.(-2)^4 + 6.(-2)^3 + 8.(-2)^2 + 3.(-2)^1 + 4.(-2)^0 mod7)


    2) Lets see whether 49140 is divisible 13 or not?
    The positive osculator of 13 is 4 (coz 13.3+1=10(4))
    We will do the following operations on the number
    4914+0(4)=4914
    491+4(4)=507
    50+7(4)=78
    7+8(4)=39
    Now since 39 is divisible by 13 so is the number itself
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  4. #4
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    It's from the definition nk+1=10 or nk-1=10

    The reason is because even numbers and 5 dont have osculators ... it's right there in the text you copied

    if n=2, then you'll note that nk+1 and nk-1 will always be odd, so they can't be divided by 10.

    Similar case for n=5
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