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What is the unit digit of A=2+2^2+2^3+...+2^10?

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What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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New post 12 Dec 2016, 08:20
2
4
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

67% (01:11) correct 33% (01:35) wrong based on 108 sessions

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What is the unit digit of \(A=2+2^2+2^3+...+2^{20}\)?

A. 0
B. 2
C. 4
D. 6
E. 8
Spoiler: OA

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Re: What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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New post 12 Dec 2016, 08:32
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\(A=2+2^2+2^3+...+2^{20}\)

Sum of GP = \(\frac{2*(2^{20} - 1)}{2 - 1} = 2*(2^{20} - 1)\)

\(20\) is a multiple of \(4\) ----> \(2^{20}\) has units digit \(6\)

\(2*(6 - 1) = 2*5 = 10\)

Units digit is \(0\)

Answer A
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Re: What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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New post 12 Dec 2016, 08:51
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nguyendinhtuong wrote:
What is the unit digit of \(A=2+2^2+2^3+...+2^{20}\)?

A. 0
B. 2
C. 4
D. 6
E. 8


\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 = 6\)
______________
Total = 0

There will be 5 such sets as \(2^4\) , \(2^8\), \(2^{12}\) , \(2^{16}\) & \(2^{20}\) ; units digit in each case will be 0

So, The sum of the units digit of the series will be 0...

Answer will be (A) 0
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Re: What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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New post 20 Dec 2016, 06:27
Fisrt, find the cyclicity of 2 = 4 (2;4;8;6)
Second, find number of cycles
Since last number of set contains 2^20, so we have 20 as greatest power. 20 is a multiple of 4, 20/4=5. So we have 5 full cycles.
Lastly, we just need to sum up all possible values of powers 2+4+8+6=20, thus 0 is the unit digit, answer A.
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What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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New post 25 Dec 2016, 17:19
Units digit pattern of 2 : 2,4,8,6
A set of 4 x 5 = 2^20
Units digit of sum of 2 to 2^20 is always 0
A
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Re: What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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Re: What is the unit digit of A=2+2^2+2^3+...+2^10?   [#permalink] 06 Feb 2019, 19:41
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