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ganeshie8 Group Title

unit's digit of \(\large 3^{3^{3^{3^{3^{3^{3}}}}}} \)

  • one month ago
  • one month ago

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  1. BSwan Group Title
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    awww to many 3's

    • one month ago
  2. BSwan Group Title
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    chooo many 3's :3

    • one month ago
  3. ganeshie8 Group Title
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    7 threes only :)

    • one month ago
  4. BSwan Group Title
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    well hmm let me think of a battern instead of calculation

    • one month ago
  5. waterineyes Group Title
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    3, 7, 1, 3, 7, 1, 3, 7, 1 this will be the last digits for every corresponding powers..

    • one month ago
  6. waterineyes Group Title
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    I think 3 will the last digit...

    • one month ago
  7. BSwan Group Title
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    3=3 mod 10 3^2= 9 mod 10 3^3=7 mod 10 3^4=1 mod 10 3^5=3 mod 10 ohk fairr enough :3 4k+q

    • one month ago
  8. ganeshie8 Group Title
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    3, 9, 7, 1...

    • one month ago
  9. waterineyes Group Title
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    9 ??

    • one month ago
  10. BSwan Group Title
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    so next step seeying 3^3^3^3^3^3 mod 4

    • one month ago
  11. waterineyes Group Title
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    Wait, let me think more, I am wrong..

    • one month ago
  12. BSwan Group Title
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    hm 3=-1 mod 4 thus 3^(odd ) =-1 mod 4 therfore checking for 4k+3 which gives 1 yeahhhh

    • one month ago
  13. BSwan Group Title
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    :3

    • one month ago
  14. ganeshie8 Group Title
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    wolfram says 7 though http://www.wolframalpha.com/input/?i=3%5E3%5E3%5E3%5E3%5E3%5E3%5E3

    • one month ago
  15. waterineyes Group Title
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    3 and 7 will repeat..

    • one month ago
  16. BSwan Group Title
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    xD

    • one month ago
  17. ganeshie8 Group Title
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    3,9,7,1 so it repeats every fourth power

    • one month ago
  18. ganeshie8 Group Title
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    how did u conclude "1" @BSwan

    • one month ago
  19. ganeshie8 Group Title
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    ``` 3=-1 mod 4 thus 3^(odd ) =-1 mod 4 therfore checking for 4k+3 ``` makes sense so far

    • one month ago
  20. ganeshie8 Group Title
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    you get 3^(4k+3) finally, right ?

    • one month ago
  21. BSwan Group Title
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    wait ahahahaha yeah yeah which gives 7 :3

    • one month ago
  22. BSwan Group Title
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    xD i only counted 0,1,2,3 insted of 1,2,3,0 xD from battern

    • one month ago
  23. BSwan Group Title
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    i love 7 anyway its hard number

    • one month ago
  24. ganeshie8 Group Title
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    Ahh that makes sense :) so basically you have worked it something like below : 3^3^3^3^3^3^3 = 3^(4k+3) = 3^3*(3^4)^k = 7(1)^k = 7 nice :)

    • one month ago
  25. BSwan Group Title
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    awww now i understand if its -1 why it keeps being -1 :o all odd powers ! dint relize that on noon , im feel im much better now lol

    • one month ago
  26. BSwan Group Title
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    hehehehe

    • one month ago
  27. BSwan Group Title
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    it does water in eyes :D

    • one month ago
  28. ganeshie8 Group Title
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    i would have added more 3's proabably haha!

    • one month ago
  29. myininaya Group Title
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    I'm seeing a 7,3,7,3,... type of pattern

    • one month ago
  30. BSwan Group Title
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    why there is water in eyes anyway ? that remindes me with a novel memories of geisha , did u read it ?

    • one month ago
  31. waterineyes Group Title
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    There are 8 3s in wolfram... But answer is yet the same..

    • one month ago
  32. ganeshie8 Group Title
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    @myininaya we need to work it from top/exponent right, pattern might be tough to predict for the last digit i thinik

    • one month ago
  33. waterineyes Group Title
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    No I have not @BSwan

    • one month ago
  34. BSwan Group Title
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    why the answer is same cuz 3=-1 mod 4 thus 3^(any odd power )=-1 mod 4

    • one month ago
  35. ganeshie8 Group Title
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    3^3^3^3^3^3^3 = 3^(3^3^3^3^3) its not same as (3^3)^3^3^3^3 or something... not so sure, need to check...

    • one month ago
  36. myininaya Group Title
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    oh you know what i was thinking about the first digit my bad

    • one month ago
  37. ganeshie8 Group Title
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    Oh! never thought of left most digit before xD last digit is easy to access using mod 10... but left most digit might be tough to work

    • one month ago
  38. BSwan Group Title
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    left most digit ?

    • one month ago
  39. myininaya Group Title
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    wait

    • one month ago
  40. BSwan Group Title
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    hmm well find a battern of (3*10)^ something hehe cool lets think of it

    • one month ago
  41. myininaya Group Title
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    isn't the first digit the ones digit and the last digit the left-most digit

    • one month ago
  42. waterineyes Group Title
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    I am no good at MOD.. we were not taught this topic in mathematics anywhere in Schooling.. :)

    • one month ago
  43. myininaya Group Title
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    i'm confused about those terms first and last

    • one month ago
  44. waterineyes Group Title
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    Last digit is one's digit..

    • one month ago
  45. ganeshie8 Group Title
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    Oh sorry, i should have used "unit's digit"

    • one month ago
  46. BSwan Group Title
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    hmm shure ?

    • one month ago
  47. ganeshie8 Group Title
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    units tens hundred thousands... yes sure, this is the correct terminologhy :)

    • one month ago
  48. BSwan Group Title
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    lavosh >.<

    • one month ago
  49. myininaya Group Title
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    well if it really was trying to figure out the units digit then it would be easy right i think \[3^3=..7\] \[3^{3^3}=(..7)^3=\] well since 7*7*7=49*7=...3 then we know \[3^{3^3}=(..7)^3=...3\] \[3^{3^{3^3}}=(...3)^3 \] 3*3*3=9*3=27 so we know \[3^{3^{3^3}}=(...3)^3=...7 \] this is how i got my 7,3,7,3,7,3... pattern

    • one month ago
  50. myininaya Group Title
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    I hope you guys know those ... just mean blah blah

    • one month ago
  51. BSwan Group Title
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    hmm ok -.- makes me feel we did nothing up there

    • one month ago
  52. BSwan Group Title
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    i thought ur gonna comment something fantacy like last digit >.< lavosh girl

    • one month ago
  53. myininaya Group Title
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    No no I wasn't trying to imply you guys were doing nothing

    • one month ago
  54. ganeshie8 Group Title
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    Now I see :) but you seem to be starting from the bottom term first ? \[ \large \color{Red}{3}^{\color{red}{3}^{3^{3^{3^{3^{3}}}}}} \]

    • one month ago
  55. myininaya Group Title
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    I was

    • one month ago
  56. myininaya Group Title
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    or I did whatever

    • one month ago
  57. ganeshie8 Group Title
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    i see haha! i think it would be bit more hard if we had to work it from the top first

    • one month ago
  58. myininaya Group Title
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    I think these type of problems aren't normally looked at the way I approached but with the whole mod thing

    • one month ago
  59. ganeshie8 Group Title
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    because 3^(3^3) gives 3^27 ****Bswan method**** (3^3)^3 gives 27^3 ****your method****

    • one month ago
  60. ganeshie8 Group Title
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    your method is easy to work because we can take mod 10 readily for the bottom 27

    • one month ago
  61. ganeshie8 Group Title
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    Bswan's method is bit more challenging as we don't know what happens in the exponent

    • one month ago
  62. BSwan Group Title
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    challenging ? hmm

    • one month ago
  63. BSwan Group Title
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    well , its not you who can juge :P

    • one month ago
  64. ganeshie8 Group Title
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    i mean, compared to (27)^3, finding the unit's digit of 3^27 is challenging

    • one month ago
  65. BSwan Group Title
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    yeah right using simple theory is much chalenging than calculating 27^3 lol redicules jugment

    • one month ago
  66. BSwan Group Title
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    what if u have already base of 27^27^27^27 ?

    • one month ago
  67. ganeshie8 Group Title
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    what i meant is, working units digit of some number like : (3408504385094385)^3 is easy compared to 3^3^3^3^3^3

    • one month ago
  68. ganeshie8 Group Title
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    since it is a single exponent, you can work it in a snap

    • one month ago
  69. BSwan Group Title
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    ur opinion , its upto what you know and what u could memorize to solve something , i wont juge in anyway both looks cool to me , but i wont say something is better cuz bla blah blah

    • one month ago
  70. ganeshie8 Group Title
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    however we cannot start from base for working the unit's digit of 3^(3^(3^(3^(3^(3^3)))))

    • one month ago
  71. myininaya Group Title
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    I don't think he was saying one way was better than the other

    • one month ago
  72. myininaya Group Title
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    just because something is found to be more challenging doesn't mean it is a worse way it means to me anyways that it is a way worth exploring

    • one month ago
  73. BSwan Group Title
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    see ,it depands on u :P

    • one month ago
  74. ganeshie8 Group Title
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    exactly ! @BSwan i was saying the problems are different, so they both require different methods. your method works for : 3^(3^(3^(3^(3^(3^3))))) myininaya's method works for : ((((((3^3)^3)^3)^3)^3)^3

    • one month ago
  75. ganeshie8 Group Title
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    they're two different problems, and two different methods. your method and the expression are challenging myininaya's method and the expression are simpler

    • one month ago
  76. BSwan Group Title
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    |dw:1408307650669:dw|

    • one month ago
  77. ganeshie8 Group Title
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    you need to work on ur vocablary a bit :P challenging is not a negative work

    • one month ago
  78. ganeshie8 Group Title
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    *word

    • one month ago
  79. BSwan Group Title
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    :*

    • one month ago
  80. ganeshie8 Group Title
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    similar problem+interesting dicussions @ http://math.stackexchange.com/questions/43327/evaluate-the-last-digit-of-77777

    • one month ago
  81. myininaya Group Title
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    I don't know why the term last digit seems weird to me but it does. @ganeshie8 It looks like that term is used other places not just here. :p I feel like a moron a little.

    • one month ago
  82. BSwan Group Title
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    ur not a moron @myininaya ;)

    • one month ago
  83. BSwan Group Title
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    only one moron could be exist ;)

    • one month ago
  84. ganeshie8 Group Title
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    i think in the arab world last digit is left most as they start writing from right side of the page xD

    • one month ago
  85. myininaya Group Title
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    The units digit is normally the last digit to be written

    • one month ago
  86. myininaya Group Title
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    I think well sometimes

    • one month ago
  87. myininaya Group Title
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    I have a good way to think about it When we say the numbers in words the last digit we say anything about in those words are the units digit

    • one month ago
  88. BSwan Group Title
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    yeah ganesh that is a problem to understand :P left / right lol

    • one month ago
  89. BSwan Group Title
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    i would love to work on last digit though its exited me for some reason

    • one month ago
  90. myininaya Group Title
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    I would like to know more about @bswan 's approach We haven't finish that way right?

    • one month ago
  91. BSwan Group Title
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    were done already :o

    • one month ago
  92. ganeshie8 Group Title
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    thats a good analogy to relate to. however unit's digit looks less controversial to me lol... because for computer science ppl and many others first/last digit makes no sense when they save numbers in queues/stacks or other data structures

    • one month ago
  93. ganeshie8 Group Title
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    I'll explain Bswan's method quick

    • one month ago
  94. ganeshie8 Group Title
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    it uses below fact \(\large 3^{4k}\) has last digit of \(\large 1\) \(k\ge 1\)

    • one month ago
  95. ganeshie8 Group Title
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    \[\large 3^{\color{Red}{3^{3^{3^{3^{3^{3}}}}}}} = 3^{\color{Red}{4k+3}} = 3^3*3^{\color{Red}{4k}} = 27*1\]

    • one month ago
  96. myininaya Group Title
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    I get that That is beautiful 4k would not work because 3^some power or even 3 is not even 4k+1 would not work either because there is no integer k such that 4k+1=3 4k+2 would not work for the same explanation as 4k 4k+3 would work because 4k+3=3 when k=0 and 4k+3=27 when k=6 and so on... Bswan, this is definitely pretty

    • one month ago
  97. BSwan Group Title
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    yeah and simply you could just say 3=-1 mod 4 :P

    • one month ago
  98. ganeshie8 Group Title
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    thats exactly how i interpreted the method also xD

    • one month ago
  99. BSwan Group Title
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    matches

    • one month ago
  100. ganeshie8 Group Title
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    thank you both xD it is some good learning to me :3

    • one month ago
  101. myininaya Group Title
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    now i have no clue how to figure out the first number (the left most) i bet that would be a killer

    • one month ago
  102. BSwan Group Title
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    well at the end of this topic , i thought of sharing this xD

    • one month ago
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  103. myininaya Group Title
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    I will pretend you are stabbing my cat because he keeps getting up in the sink after I have told him no several times

    • one month ago
  104. ganeshie8 Group Title
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    i feel it may not be possible to touch the left most digit

    • one month ago
  105. BSwan Group Title
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    no @myininaya i cant do that i love cats Ps :- i dont stab any animals as will :P i only found it funny :3

    • one month ago
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