What is the remainder of the division 2^56 by 7? : GMAT Problem Solving (PS)
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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? [#permalink]

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31 Jan 2012, 22:05
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What is the remainder of the division 2^56 by 7?

it could be discussed in past but can any one give me detailed explaanation by pattern method ?
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01 Feb 2012, 01:11
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pbull78 wrote:
what is the remainder of the divison 2power 56/7 ?
it could be discussed in past but can any one give me detailed explaanation by pattern method ?

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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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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11 Apr 2016, 04:40
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Kocket wrote:
Bunuel wrote:
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.

Could you please explain why that is?

I don´t see the logic behind this yet.

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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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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01 Feb 2012, 01:38
understood it completely thanks alot
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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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02 Feb 2012, 18:55
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
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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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02 Feb 2012, 20:04
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pappueshwar wrote:
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.
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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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21 Mar 2016, 10:36

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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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21 Mar 2016, 22:16
Chiragjordan wrote:

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No options were provided for this question in the source.
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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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11 Apr 2016, 04:32
Bunuel wrote:
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.

Could you please explain why that is?

I don´t see the logic behind this yet.
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Re: What is the remainder of the division 2^56 by 7? [#permalink]

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28 Nov 2016, 07:57
$$\frac{2^{56}}{7}$$

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

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

Remainder is $$4$$.
Re: What is the remainder of the division 2^56 by 7?   [#permalink] 28 Nov 2016, 07:57
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