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31 Aug 2020, 00:13
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Difficulty:
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71% (02:02) correct
29%
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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.
Are You Up For the Challenge: 700 Level Questions
(1) When n is divided by 4, the remainder is 1.
(2) When n is divided by 12, the remainder is 5.
Are You Up For the Challenge: 700 Level Questions
31 Aug 2020, 00:24
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IMO D
The unit digit rotation of 7^n is among the list of 7, 9, 3 and 1
Unit(7^1)=7
Unit(7^2)=9
Unit(7^3)=3
Unit(7^4)=1
Unit(7^5)=7
(1) n divides by 4 remains 1 hence the unit digit of 7^n=7 => the remainder of (7^n + 5) divides by 10 is 2
(2) n divides 12 remains 5 => n divides 4 remains 1. Same answer with (1)
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The unit digit rotation of 7^n is among the list of 7, 9, 3 and 1
Unit(7^1)=7
Unit(7^2)=9
Unit(7^3)=3
Unit(7^4)=1
Unit(7^5)=7
(1) n divides by 4 remains 1 hence the unit digit of 7^n=7 => the remainder of (7^n + 5) divides by 10 is 2
(2) n divides 12 remains 5 => n divides 4 remains 1. Same answer with (1)
Posted from my mobile device
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31 Aug 2020, 00:30
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To find: 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.
N = 1,5,9,13,17............
(unit digit of 7)
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
7^6 = 9
......
Cyclicity of 7 repeats after 4
So 7^1,5,9.... Will give 7 as unit digit. So 7^n+5 = 7+5 = 12. Remainder will be 2 when divided by 10
(Sufficient)
(2) When n is divided by 12, the remainder is 5.
N = 5,17,29,41............
(unit digit of 7)
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
7^6 = 9
......
Cyclicity of 7 repeats after 4
So 7^5,17.... Will give 7 as unit digit. So 7^n+5 = 7+5 = 12. Remainder will be 2 when divided by 10
(Sufficient)
Answer is D
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(1) When n is divided by 4, the remainder is 1.
N = 1,5,9,13,17............
(unit digit of 7)
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
7^6 = 9
......
Cyclicity of 7 repeats after 4
So 7^1,5,9.... Will give 7 as unit digit. So 7^n+5 = 7+5 = 12. Remainder will be 2 when divided by 10
(Sufficient)
(2) When n is divided by 12, the remainder is 5.
N = 5,17,29,41............
(unit digit of 7)
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
7^6 = 9
......
Cyclicity of 7 repeats after 4
So 7^5,17.... Will give 7 as unit digit. So 7^n+5 = 7+5 = 12. Remainder will be 2 when divided by 10
(Sufficient)
Answer is D
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one thing to know the last digit of number and another thing to know a whole number ... yes cyclycty repeats after 4, so what ? 
