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ganeshie8
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unit's digit of
\(\large 3^{3^{3^{3^{3^{3^{3}}}}}} \)
 one month ago
 one month ago
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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BSwan Group TitleBest ResponseYou've already chosen the best response.3
awww to many 3's
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
chooo many 3's :3
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
7 threes only :)
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
well hmm let me think of a battern instead of calculation
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
3, 7, 1, 3, 7, 1, 3, 7, 1 this will be the last digits for every corresponding powers..
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
I think 3 will the last digit...
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
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

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
3, 9, 7, 1...
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
so next step seeying 3^3^3^3^3^3 mod 4
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
Wait, let me think more, I am wrong..
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
hm 3=1 mod 4 thus 3^(odd ) =1 mod 4 therfore checking for 4k+3 which gives 1 yeahhhh
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
wolfram says 7 though http://www.wolframalpha.com/input/?i=3%5E3%5E3%5E3%5E3%5E3%5E3%5E3
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
3 and 7 will repeat..
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
3,9,7,1 so it repeats every fourth power
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
how did u conclude "1" @BSwan
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
``` 3=1 mod 4 thus 3^(odd ) =1 mod 4 therfore checking for 4k+3 ``` makes sense so far
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
you get 3^(4k+3) finally, right ?
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
wait ahahahaha yeah yeah which gives 7 :3
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
xD i only counted 0,1,2,3 insted of 1,2,3,0 xD from battern
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
i love 7 anyway its hard number
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
it does water in eyes :D
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
i would have added more 3's proabably haha!
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
I'm seeing a 7,3,7,3,... type of pattern
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
why there is water in eyes anyway ? that remindes me with a novel memories of geisha , did u read it ?
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
There are 8 3s in wolfram... But answer is yet the same..
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
@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

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
No I have not @BSwan
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
why the answer is same cuz 3=1 mod 4 thus 3^(any odd power )=1 mod 4
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

myininaya Group TitleBest ResponseYou've already chosen the best response.0
oh you know what i was thinking about the first digit my bad
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
left most digit ?
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
hmm well find a battern of (3*10)^ something hehe cool lets think of it
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
isn't the first digit the ones digit and the last digit the leftmost digit
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
I am no good at MOD.. we were not taught this topic in mathematics anywhere in Schooling.. :)
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
i'm confused about those terms first and last
 one month ago

waterineyes Group TitleBest ResponseYou've already chosen the best response.0
Last digit is one's digit..
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
Oh sorry, i should have used "unit's digit"
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
hmm shure ?
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
units tens hundred thousands... yes sure, this is the correct terminologhy :)
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

myininaya Group TitleBest ResponseYou've already chosen the best response.0
I hope you guys know those ... just mean blah blah
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
hmm ok . makes me feel we did nothing up there
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
i thought ur gonna comment something fantacy like last digit >.< lavosh girl
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
No no I wasn't trying to imply you guys were doing nothing
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

myininaya Group TitleBest ResponseYou've already chosen the best response.0
or I did whatever
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
i see haha! i think it would be bit more hard if we had to work it from the top first
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
I think these type of problems aren't normally looked at the way I approached but with the whole mod thing
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
because 3^(3^3) gives 3^27 ****Bswan method**** (3^3)^3 gives 27^3 ****your method****
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
your method is easy to work because we can take mod 10 readily for the bottom 27
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
Bswan's method is bit more challenging as we don't know what happens in the exponent
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
challenging ? hmm
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
well , its not you who can juge :P
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
i mean, compared to (27)^3, finding the unit's digit of 3^27 is challenging
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
yeah right using simple theory is much chalenging than calculating 27^3 lol redicules jugment
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
what if u have already base of 27^27^27^27 ?
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
since it is a single exponent, you can work it in a snap
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
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

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
however we cannot start from base for working the unit's digit of 3^(3^(3^(3^(3^(3^3)))))
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
I don't think he was saying one way was better than the other
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
see ,it depands on u :P
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
dw:1408307650669:dw
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
you need to work on ur vocablary a bit :P challenging is not a negative work
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
similar problem+interesting dicussions @ http://math.stackexchange.com/questions/43327/evaluatethelastdigitof77777
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
ur not a moron @myininaya ;)
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
only one moron could be exist ;)
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

myininaya Group TitleBest ResponseYou've already chosen the best response.0
The units digit is normally the last digit to be written
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
I think well sometimes
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
yeah ganesh that is a problem to understand :P left / right lol
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
i would love to work on last digit though its exited me for some reason
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
I would like to know more about @bswan 's approach We haven't finish that way right?
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
were done already :o
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
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

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
I'll explain Bswan's method quick
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
it uses below fact \(\large 3^{4k}\) has last digit of \(\large 1\) \(k\ge 1\)
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
\[\large 3^{\color{Red}{3^{3^{3^{3^{3^{3}}}}}}} = 3^{\color{Red}{4k+3}} = 3^3*3^{\color{Red}{4k}} = 27*1\]
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
yeah and simply you could just say 3=1 mod 4 :P
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
thats exactly how i interpreted the method also xD
 one month ago

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
thank you both xD it is some good learning to me :3
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

BSwan Group TitleBest ResponseYou've already chosen the best response.3
well at the end of this topic , i thought of sharing this xD
 one month ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
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

ganeshie8 Group TitleBest ResponseYou've already chosen the best response.1
i feel it may not be possible to touch the left most digit
 one month ago

BSwan Group TitleBest ResponseYou've already chosen the best response.3
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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