Thread: Power

Find the last digit of
2^3^4^5^6^7
and
2^3^5^4^5^6^7

logically and not by calculating!!

Last digit of both the equation should be 6.

Solution:

In that first equation take the first 3.
2^3 = 8
8^4 = 6
Its very easy to find the last digit of 8*8*8*8.
first multiply 8*8=16. (last digit)6*8=32 (last digit)2*8=16.

Here is the logic..whenever the last digit will come 6, it is sure at the end of any power it should be 6.

In that second equation use the same logic. 2^3^5^4 will contains the last digit 6. so finally we got 6.

i am using this logic..but i think some easiest logic might be there.

----------------
suresh

Hey i found one more logic here....

All the 2 power numbers goes to the same order like 2,4,8,6.

So multiply the numbers except 2.

3*4*5*6*7 = 2520.

Divide this number by 4. because we have a same order.

2520/4 = remainder 0. so it should be 6.

Suppose if remainder 1 will come then the answer should 2 and so on.....

Use the same logic u can get the same result for equation 2.

If anyone have other simple logic then post here....

-------------------
suresh

2^3^4^5^6^7 : last digit is 2

2^3^5^4^5^6^7 :last digit is 8


Find the correct logic..other wise i will give it later !!!

ok
2^3^4^5^.... = 2^3^(an even number) = 2^(4k +1) ...so the last digit shud be 2.

2^3^5^4.... = 2^3^(an odd number) = 2^(4k+3)... so the last digit shud be 8.

Do U hve ne questions suresh??

hi smat coder,

what you mean by k in (4k+1) ?

-----------
suresh

yes..yes...i got the logical error in your answer....

ok ....first you explain your solution (question mentioned in my previous mail)...

Then i told you where you made a mistake....

--------------------
suresh

4K+1 means any number of this form like 5,9,13....

hi smart coder,

According to the first equation 2^3^4^5^6^7 what is the value of k?
how can you find the value k ?

--------------------
suresh

i cannot find the value of k but i know that 3^(even number) is of type
4k+1 and 2^(4k+1) must end with 2..u dnt require to know the exact value of k!!

ok smart coder....i know how the value k coming....
Finally i ask one more question....

According to your logic what is the last digit value for the following equation ?
2^3^4

------------------
suresh

its 2
2^3^4 = 2^81 = 2^(4k+1)

so your answer is 2.

ok now we solve it manually.

2^3^4

if we elobarate this one we get 2^3*2^3*2^3*2^3

8*8*8*8 = 4096. so last digit is 6.

Now tell me is it your logic correct ?

---------------
suresh

ur basics are not correct..while calculating the powers u never go bottom to top..but u come top to bottom!!
2^3^4 is equal to 2^81 not 8^4 !!!

wt do u think about 2^1^2???.... tell me the last digit...4 or 2

Kindly answer me Suresh!!!

hi smart coder,

see the attached image...your question like that only...


----------------
suresh

i meant this!!!

----4
--3
2

Here i want all of our puzzle solving friends...

Because i never hear the logic "first you find top one and then finally bottom".

Waiting for our friends reply....Tell me friends which one is correct ?

------------------
suresh

lemme tell u wt i askd was based on a CAT question!!
nd do u want to challenge IIMs??

Well smart coder....
i want to clarify myself...Because "Nobody knows all the things".
That's why i need a help for other people.

Also it is very difficult to find the value for your logic...
I need a clear explantion about the 2^(4k+1).

In my next example you simply told 2^81. so you calculate the 81 like 3*3*3*3. it is single power so you can calcualte easily...if it is large then how can you find ?

If you explain this one i accept that. Waiting for your reply...

-------------------
suresh

I am not getting ur point...try to put another example!!
you want other people's view...I also agree to that..beacuse we have many intellectuals present in this forum..and I want to invite everyone to read the whole thread and then comment.
But what I asked is what I learnt at Career Launcher(a reknowned coaching for MBA) and if u ask this question to a person who is preparing for CAT will give the same answer as mine.
Thanks

wt wt wt anushya?
I m totally confused by ur question...plz write clearly!!

ok smart coder...

Explain the following one...

This is you already posted....

2^3^4^5^.... = 2^3^(an even number) = 2^(4k +1) ...so the last digit shud be 2.

2^3^5^4.... = 2^3^(an odd number) = 2^(4k+3)... so the last digit shud be 8.

In that above you mentioned 2^3^(an even number)..

how you told that is an even number and how you write 2^(4k+1).....Also how the value 1 came that place ?

------------------
suresh

Yes anushya ...the logic is defined already that u come top to bottom..
and suresh this is for you:

4^(any number) is an even number
so 3^4^(any number) = 3^(even number)

now 3^(even number) = 4k+1 type

like 3^2 = 9 = 4*2 + 1
and 3^4 = 81 = 4*20 + 1

ne other doubts?
please feel free to ask..

anushya this is for you..tell me the last digit for

2^1^2

if it is 3^(even) then 4K+1
or if 3^(odd) then 4K+3

hi smart coder,

Thanks for your all the information....This is the first time i hear these type of concept....I want to know the concept..that's why i ask so much of questions...now i have some idea to solve these type puzzle...still i have one more doubt...

you told me if it is 3^(even) then 4K+1.
suppose some one give the equation...2^7^6^9^4 or simply 7^6^9^4.

how can i solve this equation ? explain this also...

-------------------
suresh

look there can't be answers for all such type questions...u can have answers if u think very logically
for ur question:7^6^9^4 i have a solution:
6^(any number) is of form 4k except 6^1
in ur question it is obvious dat it 6^(a large number) so it will be of type 4k
Now 7^(4k) will always end with a 1.

For u anushya:
u can c many problems when brackets are not applied but d meaning is applied like the following examples:

--4
-3
2
this is not equal to

-12
2


u cannot multiply the powers until the brackets are told explicitly..

Yeah Smart coder i agree with u...

2^3^4 is equal to 2^81....

if there is no bracket then the concept is top to bottom approach... Here first we need to find 3^4 and then 2^81

if we want to find bottom to top approach then we should use bracket its mandatory...

(2^3)^5 = 2^(3+5)

Thanks
Manoj

i knew dat atleast manoj will support me..i was waiting for u only!!!

i think my this problem wud hve cleared basic mathematics concepts of many viewers!!
dats y i put this in!!

What more you want to know anushya??
what more working process??
the process is simple
just TOP to BOTTOM!!!
dats it!!
feel free to ask more!!