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Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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30 Aug 2013, 23:13

Rem(32^32^32)=Rem(4^32)^32 now 4^3 = 64 = 63+1. hence when 64 to the power anything is divided by 7, the remainder will always be 1 so, Rem(32^32)/3 = Rem(33-1)^32/3 = 1. Hence Rem(4*64^k)/7 = 4

What is the remainder when \(32^{32^{32}}\) is divided by 7?

A. 5 B. 4 C. 2 D. 0 E. 1

Please do not just post the answer, do explain as well.

I will post the Answer and the explanation after some replies.

If we use the above approach I'd work with prime as a base.

\(32^{{32}^{32}}=(28+4)^{{32}^{32}}\) now if we expand this, all terms but the last one will have 28 as a multiple and thus will be divisible by 7. The last term will be \(4^{{32}^{32}}=4^{{(2^5)}^{32}}=4^{2^{160}}=2^{2^{161}}\). So we should find the remainder when \(2^{2^{161}}\) is divided by 7.

2^1 divided by 7 yields remainder of 2; 2^2 divided by 7 yields remainder of 4; 2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2; 2^5 divided by 7 yields remainder of 4; 2^6 divided by 7 yields remainder of 1; ...

The remainder repeats the pattern of 3: 2-4-1.

So we should find \(2^{161}\) (the power of 2) is 1st, 2nd or 3rd number in the above pattern of 3. \(2^{161}\) is 2 in odd power, 2 in odd power gives remainder of 2 when divided by cyclicity number 3, so it's the second number in pattern. Which means that remainder of \(2^{2^{161}}\) divided by 7 would be the same as \(2^2\) divided by 7. \(2^2\) divided by 7 yields remainder of 4.

Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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04 Feb 2014, 21:40

Is there a post or resource about remainder cyclicities in general ? I know already about the units cyclicity but i am hoping for one about remainder cyclicities .

When we divide the power 5(32)(32) by three we get: + 2 remainder hence the remainder will be the second one in the series which is 4.
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What is the remainder when \(32^{32^{32}}\) is divided by 7?

A. 5 B. 4 C. 2 D. 0 E. 1

Please do not just post the answer, do explain as well.

I will post the Answer and the explanation after some replies.

If we use the above approach I'd work with prime as a base.

\(32^{{32}^{32}}=(28+4)^{{32}^{32}}\) now if we expand this, all terms but the last one will have 28 as a multiple and thus will be divisible by 7. The last term will be \(4^{{32}^{32}}=4^{{(2^5)}^{32}}=4^{2^{160}}=2^{2^{161}}\). So we should find the remainder when \(2^{2^{161}}\) is divided by 7.

2^1 divided by 7 yields remainder of 2; 2^2 divided by 7 yields remainder of 4; 2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2; 2^5 divided by 7 yields remainder of 4; 2^6 divided by 7 yields remainder of 1; ...

The remainder repeats the pattern of 3: 2-4-1.

So we should find \(2^{161}\) (the power of 2) is 1st, 2nd or 3rd number in the above pattern of 3. \(2^{161}\) is 2 in odd power, 2 in odd power gives remainder of 2 when divided by cyclicity number 3, so it's the second number in pattern. Which means that remainder of \(2^{2^{161}}\) divided by 7 would be the same as \(2^2\) divided by 7. \(2^2\) divided by 7 yields remainder of 4.

Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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12 Jun 2014, 21:28

Brunel,

Can we solve it by expanding 32 as (35-3) ?

Here is how I approached : 32^32^32= (35-3) ^32^32

All other terms are divisible by 7 , barring last term 3^32^32. Here is the cyclicity of

3^1 when divided by 7 gives a remainder of 3 3^2 when divided by 7 gives a remainder of 2 3^3 when divided by 7 gives a remainder of 6 3^4 when divided by 7 gives a remainder of 4 3^5 when divided by 7 gives a remainder of 5 3^6 when divided by 7 gives a remainder of 1

3^7 when divided by 7 gives a remainder of 3 3^8 when divided by 7 gives a remainder of 2 3^9 when divided by 7 gives a remainder of 6 3^10 when divided by 7 gives a remainder of 4 3^11 when divided by 7 gives a remainder of 5 3^12 when divided by 7 gives a remainder of 1 ..........

The power of 3 i.e 32^32 is equivalent to 2^160 which will be equivalent to remainder when 2^160 is divided 6

2^160 /6 = 2^159 /3 = (3-1)^159 / 3

Now, all terms in (3-1)^159 will be divisible by 3, except for the last term (-1)^159 => -1 => (-1) * 3 +2 . Therefore, the remainder is 2. This, needs to be multiplied by 2, since we had reduced 2^160/6 to 2^159/3. Therefore the final remainder is 4

Therefore, the final value comes out to be 3^4 divided 7, that gives 4 as a remainder.

I got it. Start solving from the top-most power of 32. Just imagine now 2^32 instead of 32^32. So as per cycle, 2^4 = 6. So here 2^32 = 6 as a unit digit. Now, 32^6 means 2^6, which means 2^2 = 4. So the remainder is 4.

Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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14 Nov 2014, 22:24

Bunuel wrote:

gurpreetsingh wrote:

What is the remainder when \(32^{32^{32}}\) is divided by 7?

A. 5 B. 4 C. 2 D. 0 E. 1

Please do not just post the answer, do explain as well.

I will post the Answer and the explanation after some replies.

If we use the above approach I'd work with prime as a base.

\(32^{{32}^{32}}=(28+4)^{{32}^{32}}\) now if we expand this, all terms but the last one will have 28 as a multiple and thus will be divisible by 7. The last term will be \(4^{{32}^{32}}=4^{{(2^5)}^{32}}=4^{2^{160}}=2^{2^{161}}\). So we should find the remainder when \(2^{2^{161}}\) is divided by 7.

2^1 divided by 7 yields remainder of 2; 2^2 divided by 7 yields remainder of 4; 2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2; 2^5 divided by 7 yields remainder of 4; 2^6 divided by 7 yields remainder of 1; ...

The remainder repeats the pattern of 3: 2-4-1.

So we should find \(2^{161}\) (the power of 2) is 1st, 2nd or 3rd number in the above pattern of 3. \(2^{161}\) is 2 in odd power, 2 in odd power gives remainder of 2 when divided by cyclicity number 3, so it's the second number in pattern. Which means that remainder of \(2^{2^{161}}\) divided by 7 would be the same as \(2^2\) divided by 7. \(2^2\) divided by 7 yields remainder of 4.

Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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09 Jul 2015, 11:39

Bunuel wrote:

gurpreetsingh wrote:

What is the remainder when \(32^{32^{32}}\) is divided by 7?

A. 5 B. 4 C. 2 D. 0 E. 1

Please do not just post the answer, do explain as well.

I will post the Answer and the explanation after some replies.

If we use the above approach I'd work with prime as a base.

\(32^{{32}^{32}}=(28+4)^{{32}^{32}}\) now if we expand this, all terms but the last one will have 28 as a multiple and thus will be divisible by 7. The last term will be \(4^{{32}^{32}}=4^{{(2^5)}^{32}}=4^{2^{160}}=2^{2^{161}}\). So we should find the remainder when \(2^{2^{161}}\) is divided by 7.

2^1 divided by 7 yields remainder of 2; 2^2 divided by 7 yields remainder of 4; 2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2; 2^5 divided by 7 yields remainder of 4; 2^6 divided by 7 yields remainder of 1; ...

The remainder repeats the pattern of 3: 2-4-1.

So we should find \(2^{161}\) (the power of 2) is 1st, 2nd or 3rd number in the above pattern of 3. \(2^{161}\) is 2 in odd power, 2 in odd power gives remainder of 2 when divided by cyclicity number 3, so it's the second number in pattern. Which means that remainder of \(2^{2^{161}}\) divided by 7 would be the same as \(2^2\) divided by 7. \(2^2\) divided by 7 yields remainder of 4.

I am lost here Which means that remainder of \(2^{2^{161}}\) divided by 7 would be the same as \(2^2\) divided by 7. \(2^2\) divided by 7 yields remainder of 4.

I cannot figure out why the remainder of \(2^{161}\) equals that of \(2^{2^{161}}\) when the two values are divided by 7.

Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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19 Jul 2015, 11:15

trueblue wrote:

gurpreetsingh wrote:

What is the remainder when \(32^{32^{32}}\) is divided by 7?

A. 5 B. 4 C. 2 D. 0 E. 1

Please do not just post the answer, do explain as well.

I will post the Answer and the explanation after some replies.

Is the answer B?

Intuitively, i did it like this.

32^32^32 = (28+4)^32^32 As 28 is divisible by 7, we dont need to worry about that part. Hence for the purpose of remainder, our equation boils down to 4^32^32

The cyclicity of 4 is 3 when divided by 7, hence we need to think about the value of 32^32 and what remainder it leaves when divided by 3.

Considering 32^32, it can be broken into (30+2)^32. Again 30^32 is divisible by 3. Hence we need to focus on 2^32. 2^32 can be written as (2*2)^31 = (3+1)^31. As 3^31 is also divisible by 3, we will be left with 1^31. Thus 1 would be the remainder when 32^32 is divided by 3.

This implies that 4 will be the remainder when divided by 7. Hence Answer is B.

Thank you, Karishma. Your explanations always amplify my understanding. Quick question for you - in the last part, i.e.

4*(63+1)^x Gives R4 when divided by 7.

Can you help me understand the binomial part - Binomial dictates that every term except for the last will be divisible by 7. However, can you help me understand why we are not multiplying every term we get with 4 and then trying to find the remainder?

i.e. my working after binomial is: 4*1 / 7 = R4

Why should it not be 4*(term 1/7)*(term 2/7)...*1 / 7 = Remainder

Would really appreciate if you could clarify this portion of my understanding Thank you!!

Re: What is the remainder when 32^32^32 is divided by 7? [#permalink]

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11 Feb 2016, 06:56

Bunuel wrote:

If we use the above approach I'd work with prime as a base.

\(32^{{32}^{32}}=(28+4)^{{32}^{32}}\) now if we expand this, all terms but the last one will have 28 as a multiple and thus will be divisible by 7. The last term will be \(4^{{32}^{32}}=4^{{(2^5)}^{32}}=4^{2^{160}}=2^{2^{161}}\). So we should find the remainder when \(2^{2^{161}}\) is divided by 7.

Hi Bunuel,

I fail to understand how \(4^{2^{160}}\) is equal to \(2^{2^{161}}\) Could you please shed some light on it?

Thank you!

gmatclubot

Re: What is the remainder when 32^32^32 is divided by 7?
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11 Feb 2016, 06:56

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