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# What is the remainder of (3^7^11)/5

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Math Expert
Joined: 02 Sep 2009
Posts: 52296
Re: PS-What is the remainder  [#permalink]

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08 Apr 2014, 03:42
I used the following way

3^7 = 2048. Now (2048)^11. Here last digit is 8 and cycle for this is 8,4,2,6. Now here 11th digit is 2. So when we divide we will get 2 as remainder.

First of all 3^7=2,187 not 2048.

Next 3^7^11 doe NOT equal to 2,187^11: $$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$ (if exponentiation is indicated by stacked symbols, the rule is to work from the top down).

Hope it helps.
_________________
Manager
Joined: 28 Apr 2014
Posts: 212
Re: PS-What is the remainder  [#permalink]

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30 Apr 2014, 21:16
Bunuel wrote:
LM wrote:
Helps but can you give some link or details about concept of negative remainders.

If you are preparing for the GMAT you probably shouldn't waste you valuable time on the out of the scope questions like this or on the concepts that aren't tested.

That is re-assuring because questions like these seriously make me doubt my preparation and readiness for GMAT
Intern
Joined: 06 Nov 2013
Posts: 4
Re: What is the remainder of (3^7^11)/5  [#permalink]

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14 Jun 2014, 19:32
Here's my method:

First, I found the cyclicity of the last tens and ones digit of the powers of 7 : {07, 49, 43, 01, 07, 49} therefore cyclicity = 4

7^11 with a cyclicity of 4 implies the last two digits of the power we are raising 3 to will be 43 (we only need the last two digits to determine divisibility by 4, which is also the cyclicity of 3)

We can find our units digit by finding 3^xxxx43

43/4 = 10 R3 therefore our units digit will be 3^3 which has a units digit of 7.

Senior Manager
Joined: 07 Apr 2012
Posts: 360
Re: What is the remainder of (3^7^11)/5  [#permalink]

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06 Jul 2014, 06:14
Bunuel wrote:
The last digit of 3 in positive integer power repeats in pattern of 4: {3, 9, 7, 1}. So, basically we should find the remainder upon division 7^(11) by cyclicity of 4 (to see on which number in this pattern $$7^{11}$$ falls on). $$7^{11}=(4+3)^{11}$$, now if we expand this expression all terms but the last one will have 4 in them, thus will leave no remainder upon division by 4, the last term will be $$3^{11}$$. Thus the question becomes: what is the remainder upon division $$3^{11}$$ by 4:

Hi Bunuel,

I didn't understand why we are now trying to focus on $$3^{11}$$ by 4....
We stopped looking at the rest of the question and just looking at the exponents... why is this rigth?
Math Expert
Joined: 02 Sep 2009
Posts: 52296
Re: What is the remainder of (3^7^11)/5  [#permalink]

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06 Jul 2014, 10:09
ronr34 wrote:
Bunuel wrote:
The last digit of 3 in positive integer power repeats in pattern of 4: {3, 9, 7, 1}. So, basically we should find the remainder upon division 7^(11) by cyclicity of 4 (to see on which number in this pattern $$7^{11}$$ falls on). $$7^{11}=(4+3)^{11}$$, now if we expand this expression all terms but the last one will have 4 in them, thus will leave no remainder upon division by 4, the last term will be $$3^{11}$$. Thus the question becomes: what is the remainder upon division $$3^{11}$$ by 4:

Hi Bunuel,

I didn't understand why we are now trying to focus on $$3^{11}$$ by 4....
We stopped looking at the rest of the question and just looking at the exponents... why is this rigth?

This question is out of scope of the GMAT.

I'd advice you to practice similar questions from here: special-questions-155788.html#p1341752
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Non-Human User
Joined: 09 Sep 2013
Posts: 9454
Re: What is the remainder of (3^7^11)/5  [#permalink]

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16 Jul 2018, 13:26
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Re: What is the remainder of (3^7^11)/5 &nbs [#permalink] 16 Jul 2018, 13:26

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