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# If m is a positive integer, what is the remainder when 2^m is divided

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Math Expert
Joined: 02 Sep 2009
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If m is a positive integer, what is the remainder when 2^m is divided  [#permalink]

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24 Sep 2018, 05:07
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Difficulty:

65% (hard)

Question Stats:

52% (01:12) correct 48% (01:54) wrong based on 46 sessions

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If m is a positive integer, what is the remainder when 2^m is divided by 10?

(1) m divided by 10 leaves a remainder of 0.
(2) m divided by 4 leaves a remainder of 0.

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Re: If m is a positive integer, what is the remainder when 2^m is divided  [#permalink]

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24 Sep 2018, 07:20
If m is a positive integer, what is the remainder when 2^m is divided by 10?

Cyclicity of 2^z repeats after every 4 ...
So answer would depend on div of m by 4

(1) m divided by 10 leaves a remainder of 0.
If m is 10..2^10 means 2^{4*2+2} so remainder will be same as 2^2=4
If m is 20..2^20 means 2^{4*4+4} will be same as 2^4=16, so 6
Insufficient

(2) m divided by 4 leaves a remainder of 0.
This means 2^4x will leave the same remainder as 2^4, so 6
Sufficient

B
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If m is a positive integer, what is the remainder when 2^m is divided  [#permalink]

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24 Sep 2018, 07:22
Bunuel wrote:
If m is a positive integer, what is the remainder when 2^m is divided by 10?

(1) m divided by 10 leaves a remainder of 0.
(2) m divided by 4 leaves a remainder of 0.

We are asked the units digit of $$2^m$$ . This dependent on the cyclicity of 2 . Cyclicity of 2 is 4.
1. If m is divisible by 0, as m>1, m could be 10 [in $$2^m$$ the unit digit will take the form $$2^2$$=4 as (10=2*4+2)] or 20 [in $$2^m$$ the unit digit will take the form $$2^4$$=6 as (20=5*4+0)]......will lead to varied remainders 4 or 6.............................NS
2. If m is divisible by 4, as m>1, could be 4, 8,12,....... [in $$2^m$$ the unit digit will take the form $$2^4$$=6]....will dlead to a definite remainder 6......S
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Please let me know if I am going in wrong direction.
Thanks in appreciation.

Re: If m is a positive integer, what is the remainder when 2^m is divided &nbs [#permalink] 24 Sep 2018, 07:22
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