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Thread: the largest integer that divides all the number of the form ABCABC

  1. #1
    rcs
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    the largest integer that divides all the number of the form ABCABC

    Let ABC be a 3-digit number such that its digits A, B, and C form
    an arithmetic sequence. The largest integer that divides all numbers of
    the form ABCABC is ______ ?

    thanks
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    Grand Panjandrum
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    Quote Originally Posted by rcs View Post
    Let ABC be a 3-digit number such that its digits A, B, and C form
    an arithmetic sequence. The largest integer that divides all numbers of
    the form ABCABC is ______ ?

    thanks
    So what ideas have you had for tackling this?

    Is there some related material you have covered in class?

    (As a start you will observe that ABCABC is always divisible by 1001)

    CB
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    rcs
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    some of the problems i posted are of course related in the class... i have some problems that i couldn't solve that is why i need help from here... some problems i can solve is of course not posted here because i know how. Only those i can hardly solve... i thought the more problems posted the better. im sorry sir if i have misconstrued something in here.

    thanks a lot. God Bless
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    Quote Originally Posted by rcs View Post
    some of the problems i posted are of course related in the class... i have some problems that i couldn't solve that is why i need help from here... some problems i can solve is of course not posted here because i know how. Only those i can hardly solve... i thought the more problems posted the better. im sorry sir if i have misconstrued something in here.

    thanks a lot. God Bless
    You are expected to show what you have tried or explain what exactly is the problem you are having with a particular question. Then we can help you to a solution that you will have contributed to yourself.

    We are not here to do your homework, assignments, take-out exams or provide solutions for a solutions manual for you despite what some posters here may seem to think.

    CB
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    rcs
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    yeah sir i know, and i understand that from the very first time i joined here. i cant seem think on what to do, that is why i tried to post it to have a little idea from yours or anybody from the helpers here.

    Good Day!
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    Quote Originally Posted by rcs View Post
    yeah sir i know, and i understand that from the very first time i joined here. i cant seem think on what to do, that is why i tried to post it to have a little idea from yours or anybody from the helpers here.

    Good Day!
    You have been told that every number of the form can be written $\displaystyle 1001 \times ABC$ where the digits $\displaystyle A, B, C$ are is arithmetic progression. Now try a few examples, what is the greatest common factor that you find?

    CB
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    rcs
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    1001 x 234 = 234234 , A = 2, B=3, C = 4
    1001 x ABC = ABCABC

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    Quote Originally Posted by rcs View Post
    1001 x 234 = 234234 , A = 2, B=3, C = 4
    1001 x ABC = ABCABC

    thanks
    So you have now narrowed the answer down the largest possible factor of such a number to one of $\displaystyle 1001\times a_i$ where $\displaystyle $$a_i$ is a factor of $\displaystyle 234$, but $\displaystyle 234$ is not particularly convenient for this work as it has rather more factors than is convenient.

    If instead we work with $\displaystyle A=1, B=2, C=3$ , $\displaystyle 123=41\times 3$.

    So the greatest common factor of numbers of the type being considered is one of $\displaystyle 1001, 3003, 4141, 123123$.

    So Now try $\displaystyle A=3, B=5, C=5$, $\displaystyle 345=3\times 5 \times 23$, combining this with the previous result tells us the greatest common factor is one of $\displaystyle 1001$ and $\displaystyle 3003$.

    To complete the solution it is sufficient to decide if a number of the form $\displaystyle 100\times A+10\times B+C$ with $\displaystyle A, B, C$ being in arithmetic progressions (and single digit numbers) is always divisible by $\displaystyle $$3$. So what rules do you know for testing divisibility by $\displaystyle $$3$?

    CB
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