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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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12 Dec 2016, 07:20
2
4
00:00

Difficulty:

35% (medium)

Question Stats:

65% (01:13) correct 35% (01:38) wrong based on 92 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

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

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12 Dec 2016, 07:32
1
3
$$A=2+2^2+2^3+...+2^{20}$$

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 $$0$$

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

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12 Dec 2016, 07:51
3
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...

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

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20 Dec 2016, 05:27
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.
Solved in 36 sec.
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What is the unit digit of A=2+2^2+2^3+...+2^10?  [#permalink]

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25 Dec 2016, 16:19
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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06 Feb 2019, 18:41
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Re: What is the unit digit of A=2+2^2+2^3+...+2^10?   [#permalink] 06 Feb 2019, 18:41
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