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ganeshie8
 9 months ago
unit's digit of
\(\large 3^{3^{3^{3^{3^{3^{3}}}}}} \)
ganeshie8
 9 months ago
unit's digit of \(\large 3^{3^{3^{3^{3^{3^{3}}}}}} \)

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BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3well hmm let me think of a battern instead of calculation

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.03, 7, 1, 3, 7, 1, 3, 7, 1 this will be the last digits for every corresponding powers..

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.0I think 3 will the last digit...

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.33=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

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3so next step seeying 3^3^3^3^3^3 mod 4

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.0Wait, let me think more, I am wrong..

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

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

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.03 and 7 will repeat..

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.13,9,7,1 so it repeats every fourth power

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1how did u conclude "1" @BSwan

ganeshie8
 9 months ago
Best 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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1you get 3^(4k+3) finally, right ?

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3wait ahahahaha yeah yeah which gives 7 :3

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3xD i only counted 0,1,2,3 insted of 1,2,3,0 xD from battern

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3i love 7 anyway its hard number

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1Ahh 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 :)

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3awww 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

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3it does water in eyes :D

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1i would have added more 3's proabably haha!

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I'm seeing a 7,3,7,3,... type of pattern

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

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.0There are 8 3s in wolfram... But answer is yet the same..

ganeshie8
 9 months ago
Best 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

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.0No I have not @BSwan

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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.13^3^3^3^3^3^3 = 3^(3^3^3^3^3) its not same as (3^3)^3^3^3^3 or something... not so sure, need to check...

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0oh you know what i was thinking about the first digit my bad

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1Oh! 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

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

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0isn't the first digit the ones digit and the last digit the leftmost digit

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

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0i'm confused about those terms first and last

waterineyes
 9 months ago
Best ResponseYou've already chosen the best response.0Last digit is one's digit..

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1Oh sorry, i should have used "unit's digit"

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1units tens hundred thousands... yes sure, this is the correct terminologhy :)

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0well 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

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I hope you guys know those ... just mean blah blah

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3hmm ok . makes me feel we did nothing up there

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3i thought ur gonna comment something fantacy like last digit >.< lavosh girl

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0No no I wasn't trying to imply you guys were doing nothing

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1Now I see :) but you seem to be starting from the bottom term first ? \[ \large \color{Red}{3}^{\color{red}{3}^{3^{3^{3^{3^{3}}}}}} \]

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

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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1because 3^(3^3) gives 3^27 ****Bswan method**** (3^3)^3 gives 27^3 ****your method****

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

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

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3well , its not you who can juge :P

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

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3yeah right using simple theory is much chalenging than calculating 27^3 lol redicules jugment

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3what if u have already base of 27^27^27^27 ?

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1what i meant is, working units digit of some number like : (3408504385094385)^3 is easy compared to 3^3^3^3^3^3

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1since it is a single exponent, you can work it in a snap

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3ur 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

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

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

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0just 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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1exactly ! @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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1they're two different problems, and two different methods. your method and the expression are challenging myininaya's method and the expression are simpler

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1you need to work on ur vocablary a bit :P challenging is not a negative work

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1similar problem+interesting dicussions @ http://math.stackexchange.com/questions/43327/evaluatethelastdigitof77777

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I 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.

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3ur not a moron @myininaya ;)

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3only one moron could be exist ;)

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1i think in the arab world last digit is left most as they start writing from right side of the page xD

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0The units digit is normally the last digit to be written

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I think well sometimes

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I 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

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3yeah ganesh that is a problem to understand :P left / right lol

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3i would love to work on last digit though its exited me for some reason

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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1thats 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

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1I'll explain Bswan's method quick

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

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

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I 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

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3yeah and simply you could just say 3=1 mod 4 :P

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1thats exactly how i interpreted the method also xD

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1thank you both xD it is some good learning to me :3

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0now i have no clue how to figure out the first number (the left most) i bet that would be a killer

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3well at the end of this topic , i thought of sharing this xD

myininaya
 9 months ago
Best ResponseYou've already chosen the best response.0I will pretend you are stabbing my cat because he keeps getting up in the sink after I have told him no several times

ganeshie8
 9 months ago
Best ResponseYou've already chosen the best response.1i feel it may not be possible to touch the left most digit

BSwan
 9 months ago
Best ResponseYou've already chosen the best response.3no @myininaya i cant do that i love cats Ps : i dont stab any animals as will :P i only found it funny :3
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