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

Question

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

Narenn · Accepted Answer

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

dcbark01 · Answer

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.

Narenn · Answer

I think it is D

2^86 / 9 --------> 28 times (2^3) * (2^2) / 9

Remainder when 2^3 divided by 9 is 1 and remainder when 2^2 divided by 9 is 4

So we have that 1*1*1.......28 times * 4 / 9 ------> 4/9 -------> Remainder = 4

IronHulkMan · Answer

I did it the following way:
as 9*7 = 63, 64/63 will always give a remainder of 1. 64 is 2^6.
So the new number becomes,
 /9

=> {2^(6*14)}/9 will always give remainder 1. 
2^2/9 will give e remainder of 4.
So my answer will be D(4)

nitin1negi · Answer

When a power of 2 is divided by 9, the remainder start repeating itsel after 2^6 means from 2^1 to 2^6 the remainder goes like, 2,4,8,7,5,1 and then from 2^7 it start repeating itself. so if we divide 86 by 6 it gives 2 as remainder that means only 2*2 left which give 4 as remainder for the question.

BrainLab · Answer

2^86  has a unit digit 4, 4/86 has a remainder 4 Answer (D)

ENGRTOMBA2018 · Answer

Again, you are making the mistake of going for digits to come up with the answer.

Best way will be 2^84 = 2^2*(2^3)^28 = 4*(8)^28 = 4*(9-1)^28

All the terms in (9-1)^28 will be multiples of 9 except the last one (-1)^28 = 28.

Thus, you need to find the remainder when 4*(9-1)^28 is divided by 9 or 4*(-1)^28 is divided by 9 or 4 is divided by 9, giving you a remainder of 4.

Hope this helps.

Lucky2783 · Answer

we know that when 64 is divided by 9 , remainder is 1 .

2^86 = 2^2 * 2^84 = 4 * (2^6)^14 = 4 * 64^14

Now , remember that if X= A*B then R(X/K) = ( R(A/K) *R(B/K) ) /K for eg:
65/9 remainder is 2 
65= 13*5
R(13/9) * R(5/9) = 4*5 = 20 
and R(20/9) = 2

Back to original question now - 
2^86 = 2^2 * 2^84 = 4 * (2^6)^14 = 4 * ( 64^14 )

4/9 = 4 remainder
64/9 =1 remainder

so 4 * ( 64^14 ) / 9 remainder = 4

AnswerD

FB2017 · Answer

we know cyclicity of 2 is 4. Now on dividing 86/4 we get 2 remainder for which unit's place of 2^86 should be 2^2=4. Hence dividing unit's place digit by 9 leaves remainder as 4. Option D

atjindal · Answer

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 .

epiphany1 · Answer

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

Weido · Answer

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?

generis · Answer

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

generis · Answer

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.

generis · Answer

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 Answer D *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

jabhatta2 · Answer

Bunuel ,  VeritasKarishma

Just wondering -- how to use the binomial method for the following

source : made up

(2^10)/9

KarishmaB · Answer

2^{10} = 2 * 2^9 = 2 * 8^3

= 2 * (9 - 1)^3

The remainder term when (9 - 1)^3 is divided by 9 is -1 and since all terms are multiplied by 2, the remainder is -2.
This is the same as remainder of 7.

MathRevolution · Answer

2^{86}  =  2^{84}  *  2^2  =  2^{3*28}  *  2^2

2^3  ≅ -1( when divided by 9)

(2^3)^{28}  ≅  -1^{28}  = 1

=>  \frac{1 * 4 }{ 9}

4 divided by 9 will give negative 5 as a remainder

=>  \frac{ - 5 }{ 9}

=> -5+9 = 4

Answer D

IanStewart · Answer

alitariquet is right. Naturally, citing similar questions from other prep companies is not very good evidence of what the real GMAT will test (I didn't click those links though so I don't know how similar those questions are). The lone official question linked above is about remainders by 10, not by 9, so it's about units digits. Questions about units digits are common enough on the real test, of course, and they're not analogous to the question in this thread.

That said, unrealistic questions can still be pedagogically useful (which is something everyone knows -- many test takers do drills when just starting their study, even though the GMAT never asks questions that are like those drills). The question in this thread isn't one that should ever appear in a realistic diagnostic test, but if it were used to illustrate that remainders follow cyclic patterns, say, then it could have some value.

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

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

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