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Intern  B
Joined: 21 Jul 2017
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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ronr34 wrote:
Narenn wrote:
Rule :- Expression $$\frac{A*B*C*D*E}{K}$$ will give the same remainder as the expression $$\frac{Ar*Br*Cr*Dr*Er}{K}$$ where Ar, Br, Cr, Dr, Er are the remainders when divided by K individually.

2^86 / 9 can be simplified as $$\frac{2*2*2*2.......86 times}{9}$$

2*2*2 = 8. Remainder of 8/9 is 1. We can form 28 such groups of (2*2*2) and every group will give us the remainder as 1. After forming 28 groups we are left with 2*2 which when divided by 9 will give the remainder as 4

Now apply the rule cited above

Remainder of 2^86/9 is the same as remainder of $$\frac{1*1*1*......28 times * 4}{9}$$ ----------> $$\frac{4}{9}$$ ---------> 4

Let me know if anything still unclear

I'm sorry, but isn't 8/9 yield a remainder of 8?

How do you get the number 28? I'm unclear on that.
Intern  Joined: 15 Jun 2017
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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In all the remainder of a Number with power problems, the easiest way to solve the question is by finding the cyclicity of remainders.
Remainder of 2/9 = 2
Remainder of 2^2/9 = 4
Remainder of 2^3/9 =8
Remainder of 2^4/9 =7
Remainder of 2^5/9 =5
Remainder of 2^6/9 =1
Remainder of 2^7/9 =2
Remainder of 2^8/9 =4

The remainder start repeating after 2^6 so the cyclicity is 6.
So 2^86 =2^(84+2)will have the same remainder as 2^2 when divided by 9. So the remainder is 4.
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What is the remainder when 2^86 is divided by 9?  [#permalink]

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We can also use a fermat's little theorem with a small customization as 9 is not a prime number
9 = 3 * 3
so 2^86 = 2^43 * 2^43
according to fermat's theorem
a^(p-1) = 1 mod p where a integer and p prime
so (2^2)/3=1 mod 3

2^43 * 2^43 = (2^42+1)*(2^42+1) (as we have two 3s) = so 2 * 2 = 4
Intern  Joined: 19 Aug 2017
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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Rem|2^86/9|

Now Rem|(2^3)/9| = -1

Rem|2^86/9| = Rem|(2^84 x 2^2)/9|

= Rem|[(2^3)^28 x 2^2]/9|

= (-1)^28 x 4

= 4

Final Remainder is 4
Intern  B
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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remainder when 2^86 is divided by 9
= 2^2(43)/9
= 4^43/9
= -5^43/9

now 5 has cyclicity of either 0 or 5, when power is odd , last digit will be 5
Therefore remainder=-5
since it is negative we need to add it to the divisor
therefore remainder= 9-5=4
Intern  Joined: 11 Sep 2018
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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Narenn wrote:
Rule :- Expression $$\frac{A*B*C*D*E}{K}$$ will give the same remainder as the expression $$\frac{Ar*Br*Cr*Dr*Er}{K}$$ where Ar, Br, Cr, Dr, Er are the remainders when divided by K individually.

2^86 / 9 can be simplified as $$\frac{2*2*2*2.......86 times}{9}$$

2*2*2 = 8. Remainder of 8/9 is 1. We can form 28 such groups of (2*2*2) and every group will give us the remainder as 1. After forming 28 groups we are left with 2*2 which when divided by 9 will give the remainder as 4

Now apply the rule cited above

Remainder of 2^86/9 is the same as remainder of $$\frac{1*1*1*......28 times * 4}{9}$$ ----------> $$\frac{4}{9}$$ ---------> 4

Let me know if anything still unclear

Sorry, but may I ask what is the formula for the equation, I have to solve a question similiar to that, but my question is 2^63 divided by 5.
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

I don't think that such questions will appear on the real test. Sorry, Veritis Prep be realistic.
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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My approach was to check the ending numbers of powers of 2.

2^1 ends w. 2
2^2 ends w. 4
2^3 ends w.8
2^4 ends w.6
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2^5 ends w.2
2^6 ends w.4

So we see there is a 4 pair cycle that repeats. If 2^4 ends with 8, 2^40 also ends with 8 and 2^80 too. Thus 2^84 ends with 8. 2^85 with 6 and 2^86 with 2 (beginning of new 4 pair group).

Sounds logic to you guys?
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What is the remainder when 2^86 is divided by 9?  [#permalink]

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alitariquet wrote:
I don't think that such questions will appear on the real test. Sorry, Veritis Prep be realistic.

alitariquet , I would hate for you and others who may believe the assertion above to dismiss this topic based on the belief that the topic will not be tested, only to be confronted with exactly that topic on the real test.

You may well decide that your time is better spent on other topics. Totally understandable. This similar question is from GMAT PREP 2

This similar question is from Magoosh.

This similar question is from MGMAT
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What is the remainder when 2^86 is divided by 9?  [#permalink]

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GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

Weido wrote:
My approach was to check the ending numbers of powers of 2.

2^1 ends w. 2
2^2 ends w. 4
2^3 ends w.8
2^4 ends w.6
_________________
2^5 ends w.2
2^6 ends w.4

So we see there is a 4 pair cycle that repeats. If 2^4 ends with 8, 2^40 also ends with 8 and 2^80 too. Thus 2^84 ends with 8. 2^85 with 6 and 2^86 with 2 (beginning of new 4 pair group).

Sounds logic to you guys?

Weido - Sounds logical but does not always work. In fact, using units digits cyclicity approach for remainders works every time only when the divisor is 2, 5, or 10.

Otherwise, take your good instinct one step further. Find the cyclicity of remainders.

See my post below. As mentioned, good instincts. _________________
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What is the remainder when 2^86 is divided by 9?  [#permalink]

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GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

Find the cyclicity of the remainders.

The units digit cyclicity approach works every time only when the divisor is 2, 5, or 10

People who used units digits only in this question got lucky. Here is one problem in which units digit cyclicity does NOT give the right answer.

Cyclicity of remainders
Increase 2, power by power. Divide each number by 9 and find the remainder. Continue until you see a repeating pattern.

$$2^1=2$$, and $$\frac{2}{9}=0,$$ Remainder 2
$$2^2=4$$, and $$\frac{4}{9}=0$$ Remainder 4
$$2^3=8$$, and $$\frac{8}{9}=0$$ Remainder 8
$$2^4=16$$, and $$\frac{16}{9}=1$$ Remainder 7
$$2^5=32$$, and $$\frac{32}{9}=3$$ Remainder 5
$$2^6=64$$, and $$\frac{64}{9}=7$$ Remainder 1
$$2^7=128$$, and $$\frac{128}{9}=14$$ Remainder 2
$$2^8=256$$, and $$\frac{256}{9}=28$$ Remainder 4
$$2^9=512$$, and $$\frac{512}{9}=56$$ Remainder 8

The remainder pattern started to repeat after $$2^6$$: (2, 4, 8, 7, 5, 1)

So the cyclicity of remainders is 6

Divide the exponent, 86, by cyclicity of 6.
If there is a remainder, that remainder number is the power of 2 that yields our answer.

(Example: exponent/cyclicity leaves a remainder of 5? We use 2$$^5$$ to find our answer.)

If 86/6 leaves no remainder, then we find our answer at the end of the cycle, at 2$$^6.$$
$$\frac{86}{6}=14$$ + remainder of 2, therefore:

$$2^{86}$$, when divided by 9, will have the same remainder as $$2^2$$*

From above, when $$2^2$$ is divided by 9,
the remainder is 4

*Divide the exponent by the cycle number to figure out where, in the cycle of 6, the exponent would fall.
(6*14)=84, so 2$$^{84}$$ would "hit" the cycle in the same position as 2$$^6$$ (84 is a multiple of 6).
Start the cycle of 6 over.
2$$^{85}$$ = same remainder as 2$$^1$$, and
2$$^{86}$$ = same remainder as 2$$^2$$
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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Hi Bunuel - do you mind explaining for me when we should know to break down 2^X and when we shouldn't?

For example - in this question, we break down 2^86 / 9 into (9-2^3)... but in an earlier question (333^222)/7 -- we narrow it down to 2^444/7, without breaking it down further.

Do you understand my question?
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

$$2^{86} = 2^{84}*2^2 = (2^3)^{28}*4 = 8^{28}*4$$
= $$(9 - 1)^{28}*4$$
= (9L + 1)*4, for any positive integer L
= 36L + 4

When divided by 9
--> (36L + 4)/9 = 4L + 4/9
Remainder = 4

IMO Option D

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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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dcbark01 wrote:
I used the Binomial Theory (which I admit I'm a little shaky on...)

1. We need to find a relationship between our dividend (2^86) and divisor (9)

2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28

We can expand this into binomial form as:

4*(9-1)^28

Every term will have a factor of 9 in it EXCEPT the last term (-1)^28 which is just equal to 1. Don't forget to distribute the 4 out, and voila, we have our remainder of 4*(1) = 4 therefore answer D.

In this what happens if the power is odd i.e. (-1)^27 (say).... will the remainder then be 4*-1/9 i.e. 5?
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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dcbark01 wrote:
I used the Binomial Theory (which I admit I'm a little shaky on...)

1. We need to find a relationship between our dividend (2^86) and divisor (9)

2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28

We can expand this into binomial form as:

4*(9-1)^28

Every term will have a factor of 9 in it EXCEPT the last term (-1)^28 which is just equal to 1. Don't forget to distribute the 4 out, and voila, we have our remainder of 4*(1) = 4 therefore answer D.

In this what happens if the power is odd i.e. (-1)^27 (say).... will the remainder then be 4*-1/9 i.e. 5?

Yes. The remainder will be 5 if the power is odd.

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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

When a power of 2 is divided by 9, the remainder start repeating itsel after 2^6:

2^1 => remainder = 2
2^2 => remainder = 4
2^3 => remainder = 8
2^4 => remainder = 7
2^5 => remainder = 5
2^6 => remainder = 1

2^7 => remainder = 2

Therefore,
89/6 gives 2 as reminder => 2^100 and 2^2 have the same remainder when divided by 9, ie 4.
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Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

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Always opt for pattern recognition when complex math is involved.
Remainder
2/9 2
4/9 4
8/9 8
16/9 7
32/9 5
64/9 1
128/9 2 - Pattern is repeating
If you repeat upto 2^86 or find a shortcut with given pattern, you will find yourself with 4 as the remainder for 2^86/9. Re: What is the remainder when 2^86 is divided by 9?   [#permalink] 05 Sep 2019, 23:38

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