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

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.
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understood it completely thanks alot

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

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.

BRO....OPTIONS PLEASE?

Please Post the Options too
Peace Out
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No options were provided for this question in the source.
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Could you please explain why that is?

I don´t see the logic behind this yet.

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

\(2^3 = 1 (mod 7)\)

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

Remainder is \(4\).

[quote="pbull78"]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