What is the remainder when 8^643/132? : GMAT Problem Solving (PS)
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# What is the remainder when 8^643/132?

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What is the remainder when 8^643/132? [#permalink]

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16 Oct 2012, 13:20
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What is the remainder when 8^643/132?

A. 116
B. 117
C. 120
D. 109

[Reveal] Spoiler:
8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

[Reveal] Spoiler: OA

Last edited by Bunuel on 17 Oct 2012, 03:29, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 14:32
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$

$$33=32+1=2^5+1$$

$$2^{1929}=(2^5)^{385}\cdot2^4=(33-1)^{385}\cdot{16}=(M33-1)\cdot{16}=M33-16=M33-33+17=M33+17$$.
($$M33$$ designates multiple of $$33$$.)

Hence, the remainder is $$17$$.
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Last edited by EvaJager on 16 Oct 2012, 14:39, edited 1 time in total.
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 14:38
EvaJager wrote:
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$\frac{2^{1929}}{2^2\cdot{3}\cdot{11}}=\frac{2^{1927}}{3\cdot{11}}$$

So, we have to find the remainder when $$2^{1927}$$ is divided by $$33$$.
$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.
($$M33$$ designates multiple of $$33$$.)

Hence, the remainder is 29.

Thanks EvaJager.

The options were

a. 116
b. 117
c. 120
d. 109

116 is highlighed as the answer. How to arrive at this?
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 14:41
g3kr wrote:
EvaJager wrote:
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$\frac{2^{1929}}{2^2\cdot{3}\cdot{11}}=\frac{2^{1927}}{3\cdot{11}}$$

So, we have to find the remainder when $$2^{1927}$$ is divided by $$33$$.
$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.
($$M33$$ designates multiple of $$33$$.)

Hence, the remainder is 29.

Thanks EvaJager.

The options were

a. 116
b. 117
c. 120
d. 109

116 is highlighed as the answer. How to arrive at this?

My previous posts are not correct...
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 14:51
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g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.
($$M33$$ designates multiple of $$33$$.)

$$2^{1929}=2^{1927}\cdot2^2=(M33+29)\cdot{4}=M132+29\cdot{4}=M132+116$$.
$$M33+29$$ means $$33n+29$$, for some integer $$n$$. Then $$(M33+29)\cdot{4}=132n+29\cdot{4}=M132+116$$.

Hence, the remainder is 116.
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 15:54
EvaJager wrote:
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.

($$M33$$ designates multiple of $$33$$.)

$$2^{1929}=2^{1927}\cdot2^2=(M33+29)\cdot{4}=M132+29\cdot{4}=M132+116$$.
$$M33+29$$ means $$33n+29$$, for some integer $$n$$. Then $$(M33+29)\cdot{4}=132n+29\cdot{4}=M132+116$$.

Hence, the remainder is 116.

What is wrong if i cancel out 2^2 in the numerator and divide by 33 . I still have to get the same reminder right?
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 16:22
EvaJager wrote:
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.
($$M33$$ designates multiple of $$33$$.)

$$2^{1929}=2^{1927}\cdot2^2=(M33+29)\cdot{4}=M132+29\cdot{4}=M132+116$$.
$$M33+29$$ means $$33n+29$$, for some integer $$n$$. Then $$(M33+29)\cdot{4}=132n+29\cdot{4}=M132+116$$.

Hence, the remainder is 116.

Is there some other method of doing it?
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 17:51
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g3kr wrote:
EvaJager wrote:
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.

($$M33$$ designates multiple of $$33$$.)

$$2^{1929}=2^{1927}\cdot2^2=(M33+29)\cdot{4}=M132+29\cdot{4}=M132+116$$.
$$M33+29$$ means $$33n+29$$, for some integer $$n$$. Then $$(M33+29)\cdot{4}=132n+29\cdot{4}=M132+116$$.

Hence, the remainder is 116.

What is wrong if i cancel out 2^2 in the numerator and divide by 33 . I still have to get the same reminder right?

Hi g3kr,

Both of the last 2 solutions fetching 29 and 116 are correct processwise, but for a logical mistake in former.

Consider the case, 16/10 will have a remainder of 6, but its not same as 8/5 which has a remainder of 3 only.

The difference? notice that for any y/x remainder can not be greater than x, hence if you have factored out a number from numerator and denominator, you have reduced remainder by that times.
In our example of 16/10: if we make it 8/5 by dividing each term by 2 and get 3 as remainder. we should multiply result by 2 to get correct ans 6.

hence, once you arrived in first solution (M33+29)

you should remember to multiply it by 4, the one which was factored out earlier. Or better dont factor out and cancel a term in a remainder problem.

Hope it helps.
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Re: Remainder problem. [#permalink]

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16 Oct 2012, 18:06
Thank you so much
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16 Oct 2012, 18:26
g3kr wrote:
Thank you so much

You are welcome.
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16 Oct 2012, 19:39
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What is wrong if i cancel out 2^2 in the numerator and divide by 33 . I still have to get the same reminder right?[/quote]

Hi g3kr,

Both of the last 2 solutions fetching 29 and 116 are correct processwise, but for a logical mistake in former.

Consider the case, 16/10 will have a remainder of 6, but its not same as 8/5 which has a remainder of 3 only.

The difference? notice that for any y/x remainder can not be greater than x, hence if you have factored out a number from numerator and denominator, you have reduced remainder by that times.
In our example of 16/10: if we make it 8/5 by dividing each term by 2 and get 3 as remainder. we should multiply result by 2 to get correct ans 6.

hence, once you arrived in first solution (M33+29)

you should remember to multiply it by 4, the one which was factored out earlier. Or better dont factor out and cancel a term in a remainder problem.

Hope it helps.[/quote]

Thanks,

Even I had the same ambiguity..
But just to be sure.. there wont be both options in the answer : 29 & 116 ?
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16 Oct 2012, 20:33
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mindmind wrote:
Even I had the same ambiguity..
But just to be sure.. there wont be both options in the answer : 29 & 116 ?

Both 29 and 116 can be in the options because the answer is only 116. In fact, it is quite likely that both will be there since if there is a way to trip you, GMAT will not let it go. You can choose to cancel off terms in the numerator and denominator or you can choose to keep them as it is. In either case, answer will be 116.

$$\frac{8^{643}}{132} = \frac{8*8^{642}}{132}$$

We get $$2*\frac{8^{642}}{33} = 2*\frac{2^{1926}}{33}$$

Now, notice that $$2^5 = 32$$ is 1 less than 33 so we use it. Why? (Check out this post: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/ )
1926 is 1 more than a multiple of 5 so we keep aside a 2.

$$2*2*\frac{2^{1925}}{33} = 4*\frac{32^{385}}{33}$$

We get $$4*\frac{(33 - 1)^{385}}{33}$$

Using binomial theorem (discussed in the link given above), we see that the remainder is -4 i.e. 29.

Recall that we cancelled off 4 from the numerator in the beginning. We need to multiply it back so actual remainder is 29*4 = 116
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16 Oct 2012, 20:45
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mindmind wrote:
What is wrong if i cancel out 2^2 in the numerator and divide by 33 . I still have to get the same reminder right?

Hi g3kr,

Both of the last 2 solutions fetching 29 and 116 are correct processwise, but for a logical mistake in former.

Consider the case, 16/10 will have a remainder of 6, but its not same as 8/5 which has a remainder of 3 only.

The difference? notice that for any y/x remainder can not be greater than x, hence if you have factored out a number from numerator and denominator, you have reduced remainder by that times.
In our example of 16/10: if we make it 8/5 by dividing each term by 2 and get 3 as remainder. we should multiply result by 2 to get correct ans 6.

hence, once you arrived in first solution (M33+29)

you should remember to multiply it by 4, the one which was factored out earlier. Or better dont factor out and cancel a term in a remainder problem.

Hope it helps.

Thanks,

Even I had the same ambiguity..
But just to be sure.. there wont be both options in the answer : 29 & 116 ?

Hi mindmind (too many minds :D),

Exactly what I explained with example of 16/10 and 8/5. if i ask you remainder of 16/10 and give u option of 6 and 3, will you be confused? I dont think so

If u factor out a number, the "fraction" remains same but "remainder" does not. please re-read the post. Also this is why it is important not to factor out in remainder problem.
Also try to check one more way of understanding it:
a remainder is shown as below:
if n = qm +r
then kn = kqm +kr
where Kn is a multiple of n and km is multiple of m. But this changes remainder from r to kr.

Hope it is clear..
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27 Oct 2014, 11:26
Is there a trick to this question?
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Re: What is the remainder when 8^643/132? [#permalink]

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17 Mar 2016, 23:06
EvaJager wrote:
g3kr wrote:

what is the remainder when 8^643/132?

8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

$$8^{643}=(2^3)^{643}=2^{1929}$$
$$132=2^2\cdot{3}\cdot{11}$$

$$33=32+1=2^5+1$$

$$2^{1927}=(2^5)^{385}\cdot2^2=(33-1)^{385}\cdot{4}=(M33-1)\cdot{4}=M33-4=M33-33+29=M33+29$$.
($$M33$$ designates multiple of $$33$$.)

$$2^{1929}=2^{1927}\cdot2^2=(M33+29)\cdot{4}=M132+29\cdot{4}=M132+116$$.
$$M33+29$$ means $$33n+29$$, for some integer $$n$$. Then $$(M33+29)\cdot{4}=132n+29\cdot{4}=M132+116$$.

Hence, the remainder is 116.

I got the correct answer but took me very long to do it, which would be tricky in exam conditions. is this really a gmat question??
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What is the remainder when 8^643/132? [#permalink]

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01 Dec 2016, 06:39
g3kr wrote:
What is the remainder when 8^643/132?

A. 116
B. 117
C. 120
D. 109

[Reveal] Spoiler:
8^643 = (2^3)^643=2^1929

2^x has a cyclicity of 3 , 1929 is divisible by 3..

We can solve this question very quickly if we apply Euler’s theorem.

$$132 = 2^2*3*11$$

We can't apply theorem yet, because 8 and 132 are not co-prime.

$$\frac{8^{642}*2^3}{2^2*33} = \frac{8^{642}*2}{33}$$

$$Ɵ(33) = 33 * \frac{2}{3} * \frac{10}{11} = 20$$

Hence: $$8^{20} = 1 (mod 33)$$

$$\frac{8^{640}*8^2*2}{33} = \frac{(8^{20})^{32}*64*2}{33} = \frac{1 * (-2)*2}{33} = -4 (mod 33) = 29 mod (33)$$

Now we multiply by cancelled factor $$4$$

$$29*4 (mod 33*4) = 116 (mod 132)$$

Our remainder is $$116$$.

What is the remainder when 8^643/132?   [#permalink] 01 Dec 2016, 06:39
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