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Director  Joined: 16 May 2014
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GRE Math Challenge-units digit of 57^n [#permalink]
Expert's post 00:00

Question Stats: 84% (00:24) correct 15% (00:26) wrong based on 13 sessions
Which of the following could be the units digit of $$57^n$$ where n is a positive integer?

Indicate all such digits.

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
(F) 5
(G) 6
(H) 7
(I) 8
(J) 9

[Reveal] Spoiler: OA
B,D,H,J
[Reveal] Spoiler: OA

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If you find this post helpful, please press the kudos button to let me know !  Intern Joined: 18 Jun 2014
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(B) 1
(D) 3
(H) 7
(J) 9
Intern Joined: 10 Jun 2018
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Re: GRE Math Challenge-units digit of 57^n [#permalink]
As per me,we should get the answer from multiple of 7 .
In which, all of the options are answer. Intern Joined: 08 Oct 2018
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Kudos [?]: 8  , given: 2

Re: GRE Math Challenge-units digit of 57^n [#permalink]
2
KUDOS
Easiest to just take the units number 7 from 57 and then test exponents of this. 7^1 = 7, 7^2 = 49, 7^3 = 343, 7^4 = 2401, 7^5 =16807.

Once you get to 7^5, you notice the pattern for the units, 7...9...3...1...7. So regardless of the value of the exponent, the result for the units will be one of those 4 values.

The answers are 7, 9, 3, 1.

(You can test this by raising 57^2 and getting the answer 2349, which has the same tens and units number as 7^2. The same is for 57^3 = 185193)
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Re: GRE Math Challenge-units digit of 57^n [#permalink]
Expert's post
NixonDutt wrote:
As per me,we should get the answer from multiple of 7 .
In which, all of the options are answer.

Hi
power $$x^2$$ means x multiplied two times x*x
So $$57^n$$ means 57*57*57...n times

Each digit has a cyclicity when multiplied x times
So 7 has cyclicity which repeats after every 4th term
7^1=7..7^2=7*7=9....7^3=7*7*7=9*7=3...7^4=7*7*7*7=3*7=1
So cyclicity is 7,9,3,1,7....

Also it is multiple of odd, so all even are out..
5 as units digit requires 5 as a multiple but here we have 7..
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html Re: GRE Math Challenge-units digit of 57^n   [#permalink] 28 Oct 2018, 02:06
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