small
Lv 7
small asked in Science & MathematicsMathematics · 8 months ago

How to determine if a large number (say in quintillion, 19 digits) is perfectly divisible by 999 without using the process of division?

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    999999 is divisible by 999

    999999999 is divisible by 999

    In general

    10^(3t) - 1 is divisible by 999

    So let's say you have a number like 1,378,492,265,446,978,132.  Is it divisible by 999?  Well, we know that 999,999,999,999,999,999 is divisible by 999, so we can start there

    1,378,492,265,446,978,132 - 999,999,999,999,999,999 =>

    378,492,265,446,978,133

    999,999,999,999,999 * 378 = 378,000,000,000,000 - 378

    378,492,265,446,978,133 - (378,000,000,000,000 - 378) =>

    492,265,446,978,133 + 378 =>

    492,265,446,978,511

    999,999,999,999 * 492 = 492,000,000,000,000 - 492

    265,446,978,511 + 492 = 265,446,979,003

    Do you see a pattern emerging?

    1 , 378 , 492 , 265 , 446 , 978 , 132

    Break it up and add each 3 digit section

    1 + 378 + 492 + 265 + 446 + 978 + 132 =>

    379 + 757 + 1424 + 132 =>

    2692

    2 , 692

    692 + 2 = 694

    694 is not divisible by 999, so 1,378,492,265,446,978,132 is not divisible by 999.

    Let's use a number that is divisible by 999 and see if it works:

    ‭1 , 488 , 038 , 328 , 957 , 537 , 648‬

    1 + 488 + 038 + 328 + 957 + 537 + 648 = 2 , 997

    2 + 997 = 999

    Looks like it works.

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  • geezer
    Lv 7
    8 months ago

    Add up the digits .. if that is divisable by 3 then the number is divisable by 999

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  • JOHN
    Lv 7
    8 months ago

    Hope the below is readable

    Attachment image
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  • ?
    Lv 7
    8 months ago

    If the sum of the digits are divisible by 3, then the large number is divisible by 999.

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  • small
    Lv 7
    8 months ago

    I have a simple process which uses only the addition process to get the answer and it would also determine the remnant if the large number is not perfectly divisible by 999.... and even the process of addition needs to be applied not more than twice.

    • billrussell42
      Lv 7
      8 months agoReport

      so you are answering your own question?

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