What is the remainder of the division 2^56 by 7?

Question

A. 1
B. 2
C. 3
D. 4
E. 5

Bunuel · Accepted Answer

Such kind of remainder questions are almost always about the pattern recognition. Now, let's check: 2^1 divided by 7 yields the remainder of 2; 2^2 divided by 7 yields the remainder of 4; 2^3 divided by 7 yields the remainder of 1; 2^4 divided by 7 yields the remainder of 2; 2^5 divided by 7 yields the remainder of 4; 2^6 divided by 7 yields the remainder of 1; ... As you can see the remainder repeats in pattern of 3: {2, 4, 1}. 2^56 will have the same reminder as 2^2 (divide 56 by 3 and find the remainder: 56=3*18+2, so it'll match with 2^2). Hence the remainder will be 4. If it were 2^60 instead of 2^56 then as 60 has the remainder of 0 upon division by 3, then 2^60 would yield the same remainder as 2^3, so 1. Questions to practice: PS questions on remainders: search.php?search_id=tag&tag_id=199 DS questions on remainders: search.php?search_id=tag&tag_id=198 Also check theory on remainders: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html Hope it helps.

pbull78 · Answer

understood it completely thanks alot

pappueshwar · Answer

hi,
if 56 is divided by 3, i get the reminder as 2, then the answer should be 2 right? how are u getting 4 as reminder and taking that as answer? request to let me know

Dreaming · Answer

thats because we are not asked for the remainder of 56 divided by 3. we are asked for the remainder of 2^56 divided by 7
56/3=2 indicates that is is the 2nd 'element' in the set ....

2^55 divided by 7 yields the remainder of 2;
 2^56 divided by 7 yields the remainder of 4; 2^57 divided by 7 yields the remainder of 1;

...which as u see from the reasoning given by brunnel....gives a remainder as 4.

stonecold · Answer

BRO....OPTIONS PLEASE?

Please Post the Options too
Peace Out

Bunuel · Answer

No options were provided for this question in the source.

Kocket · Answer

Could you please explain why that is?

I don´t see the logic behind this yet.

Bunuel · Answer

Check the links below:
 Cyclicity on the GMAT 
 Divisibility and Remainders on the GMAT

Theory on remainders problems 
 Tips on remainders

Units digits, exponents, remainders problems

vitaliyGMAT · Answer

\frac{2^{56}}{7}

2^3 = 1 (mod 7)

\frac{(2^3)^{18}*2^2}{7}  =  \frac{1*4}{7}

Remainder is  4 .

sidharth2412 · Answer

What is the remainder of the division 2^56 by 7?

Using Euler method of finding remainders
2^56 mod 7
E(7)=6
As 2 and 7 are co-prime, 2^6k mod 7=1
So, 2^56 mod 7= 2^54 * 2^2 mod 7= 4 mod 7= 4

Ans. is 4

BrushMyQuant · Answer

We need to find the the remainder when  2^{56}  is divided by 7 ?

Let's solve the problem using two Methods:

Method 1: Cyclicity of Remainder of power of 2 by 7

To solve this problem we need to find the cycle of remainder of power of 2 when divided by 7

Remainder of  2^1  (=2) by 7 = 2
Remainder of  2^2  (=4) by 7 = 4 
Remainder of  2^3  (=8) by 7 = 1
Remainder of  2^4  (=16) by 7 =   2   
Remainder of  2^5  (=32) by 7 =   4  
Remainder of  2^6  (=64) by 7 =   1

=> Cycle is 3

Remainder of 56 by 3 = 2
=> Remainder of  2^{56}  by 7 = Remainder of 2^2 by 7 =4

Method 2: Binomial Theorem

2^{56}  =  2^{54+2}  = 2^54 * 2^2 = 2^(3*18) * 4 = 8^18 * 4

Remainder of  2^{56}  by 7 = Remainder of 8^18 by 7 * Remainder of 4 by 7 = (Remainder of 8^18) * 4

Remainder of 8^18 = Remainder of (7 + 1)^18 by 7
 
 Now, if we use Binomial Theorem to expand this then all the terms except the last term will be a multiple of 7
=> All terms except the last term will give remainder of 0 when divided by 7 
=> Problem is reduced to what is the remainder when the last term (i.e. 18C18 * 7^0 * 1^18) is divided by 7
=> Remainder of 1 * 1 * 1 is divided by 7
=> Remainder of 1 divided by 7 = 1
=> Remainder of  2^{56}  by 7 = (Remainder of 8^18) by 7 * 4 = 1 * 4 = 4

So,  Answer will be D 
Hope it Helps!

Watch following video to MASTER Remainders by 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

bumpbot · Answer

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What is the remainder of the division 2^56 by 7?

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What is the remainder of the division 2^56 by 7?

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