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Math Help - 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 1001 \times ABC where the digits 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

    thanks
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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 1001\times a_i where $$a_i is a factor of 234, but 234 is not particularly convenient for this work as it has rather more factors than is convenient.

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

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

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

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

    CB
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