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# If n is a positive integer, what is the remainder when (7^n + 5)

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Manager
Joined: 03 Nov 2019
Posts: 54
If n is a positive integer, what is the remainder when (7^n + 5)  [#permalink]

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11 Dec 2019, 09:17
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Difficulty:

45% (medium)

Question Stats:

73% (02:09) correct 27% (02:31) wrong based on 33 sessions

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If n is a positive integer, what is the remainder when (7^n + 5) is divided by 10?

(1) When n is divided by 4, the remainder is 1.

(2) When n is divided by 12, the remainder is 5.
Director
Status: Manager
Joined: 27 Oct 2018
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Location: Egypt
GPA: 3.67
WE: Pharmaceuticals (Health Care)
Re: If n is a positive integer, what is the remainder when (7^n + 5)  [#permalink]

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11 Dec 2019, 13:03
As the cyclicity of "7" is of 4 numbers, the question is actually asking about the remainder when dividing n by 4.

from statement (1), n = 4x+1, which means that n can be equal to (1,5,9,13,...)
so all the possible values of n gives 1 when divided by 4 --> sufficient

from statement (2), n = 12x+5, which means that n can be equal to (5,17,29,41,...)
so all the possible values of n gives 1 when divided by 4 --> sufficient

D

as the remainder of n when divided by 4 is 1, the unit digit of $$7^n$$ is always 7,
and the unit digit of $$7^n+5$$ is always 2 --> which is the answer for the question
Re: If n is a positive integer, what is the remainder when (7^n + 5)   [#permalink] 11 Dec 2019, 13:03
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