• @[email protected]
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    8717 days ago

    The divisability rule for 7 is that the difference of doubled last digit of a number and the remaining part of that number is divisible by 7.

    E.g. 299’999 → 29’999 - 18 = 29’981 → 2’998 - 2 = 2’996 → 299 - 12 = 287 → 28 - 14 = 14 → 14 mod 7 = 0.

    It’s a very nasty divisibility rule. The one for 13 works in the same way, but instead of multiplying by 2, you multiply by 4. There are actually a couple of well-known rules for that, but these are the easiest to remember IMO.

    • @[email protected]
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      1317 days ago

      If all of the digits summed recursively reduce to a 9, then the number is divisible by 9 and also by 3.

      If the difference between the sums of alternating sets digits in a number is divisible by 11, then the number itself is divisible by 11.

      That’s all I can remember, but yay for math right?

      • @[email protected]
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        817 days ago

        Well, on the side of easy ones there is “if the last digit is divisible by 2, whole number is divisible by 2”. Also works for 5. And if you take last 2 digits, it works for 4. And the legendary “if it ends with 0, it’s divisible by 10”.

      • @levzzz
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        116 days ago

        The 9 rule works for 3 too The 6 rule is if (divisible by 3 and divisible by 2)

    • @candybrie
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      516 days ago

      I think it might be easier just to do the division.

  • @[email protected]
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    17 days ago

    ⅐ = 0.1̅4̅2̅8̅5̅7̅

    The above is 42857 * 7, but you also get interesting numbers for other subsets:

         7 * 7 =     49
        57 * 7 =    399
       857 * 7 =   5999
      2857 * 7 =  19999
     42857 * 7 = 299999
    142857 * 7 = 999999
    

    Related to cyclic numbers:

    142857 * 1 = 142857
    142857 * 2 = 285714
    142857 * 3 = 428571
    142857 * 4 = 571428
    142857 * 5 = 714285
    142857 * 6 = 857142
    142857 * 7 = 999999
    
  • @[email protected]
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    17 days ago

    Never realized there are so many rules for divisibility. This post fits in this category:

    Forming an alternating sum of blocks of three from right to left gives a multiple of 7

    299,999 would be 999 - 299 = 700 which is divisible by 7. And if we simply swap grouped digits to 999,299, it is also divisible by 7 since 299 - 999 = -700.

    And as for 13:

    Form the alternating sum of blocks of three from right to left. The result must be divisible by 13

    So we have 999 - 999 + 299 = 299.

    You can continue with other rules so we can then take this

    Add 4 times the last digit to the rest. The result must be divisible by 13.

    So for 299 it’s 29 + 9 * 4 = 65 which divides by 13. Pretty cool.

    • Grubberfly 🔮
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      16 days ago

      That is indeed an absurd amount of rules (specially for 7) !

      It should be fun to develop each proof. Particularly the 1,3,2,-1,-3,-2 rule, which at first sight seems could be easily expanded to any other number.

    • @Klear
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      16 days ago

      Wait until you learn of 51/17.

      • @prime_number_314159
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        216 days ago

        With 17, I understand that you’re referring to how 299,999 is also divisible by 17. What is the 51 reference, though? I know there’s 3,999,999,999,999 but that starts with a 3. Not the same at all.

        • @Klear
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          216 days ago

          57 / 17 = 3. That messes up with my brain.

          • @scutiger
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            416 days ago

            Your math is wrong.

            • @Klear
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              216 days ago

              That it is. And it’s not just my math that is wrong.

  • @[email protected]
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    816 days ago

    Can we just say it isn’t? Like that’s an exception, and then the rest of math can just go on like normal

  • @Etterra
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    416 days ago

    So what? Being a prime number doesn’t mean it can’t be a divisor. Or is it the string of 9s that’s supposed to be upsetting? Why? What difference does it make?

  • @[email protected]
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    416 days ago

    Phew, for a moment I worried that 2.9999… was divisible by 7 and I woke up in some kind of alternate universe

    • @Shard
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      417 days ago

      Yes technically almost every number is divisible by another in some way and you’re left with a remainder that spans plenty of decimal places.

      But common parlance when something is said to be divisible is that the end results is a round number…

  • m1o9n6s8t8e3r
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    117 days ago

    @m4m4m4m4

    Caught me… A pretty quick way to see this and the title is using Fermat’s little theorem which states that k^(p-1) ~ 1 mod p for nonzero k. Using this we can write

    3*10^5 ~ 3^6 ~ 1 mod 7
    and
    3*10^8 ~ 3*(-3)^8 ~ 3^9 ~ 16^9 ~ 2^36 ~ (212)3 ~ 1^3 ~ 1 mod 13