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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.
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(1) When n is divided by 4, the remainder is 1.
(2) When n is divided by 12, the remainder is 5.
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Posts: 79
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)
Posted from my mobile device
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
yashikaaggarwal
Senior Moderator - Masters Forum
Joined: 18 Jan 2020
Last visit: 10 Aug 2026
Posts: 3,095
Location: India
Schools: Cranfield (A) Warwick MiM "23 (M)
GPA: 4
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Schools: Cranfield (A) Warwick MiM "23 (M)
Posts: 3,095
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
Posted from my mobile device
_________________
(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
Posted from my mobile device
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Posts: 997
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.
\(n=4x+1\)
\(7^(4x+1) + 5\)
unit digit of this number will be \(7+5=12 \)
we get the remainder \(2\) when divided by \(10\)
(2) When n is divided by 12, the remainder is 5.
\(n=12x+5\)
\(7^(12x+5) + 5\)
unit digit of this number will be \(7+5=12 \)
we get the remainder \(2\) when divided by \(10\)
So D
(1) When n is divided by 4, the remainder is 1.
\(n=4x+1\)
\(7^(4x+1) + 5\)
unit digit of this number will be \(7+5=12 \)
we get the remainder \(2\) when divided by \(10\)
(2) When n is divided by 12, the remainder is 5.
\(n=12x+5\)
\(7^(12x+5) + 5\)
unit digit of this number will be \(7+5=12 \)
we get the remainder \(2\) when divided by \(10\)
So D
yashikaaggarwal
yashikaaggarwal what is the relation between unit digit cyclicty and finding of remainder ?
one thing to know the last digit of number and another thing to know a whole number ... yes cyclycty repeats after 4, so what ? dammit still have questions with quant
yashikaaggarwal
Senior Moderator - Masters Forum
Joined: 18 Jan 2020
Last visit: 10 Aug 2026
Posts: 3,095
Location: India
Schools: Cranfield (A) Warwick MiM "23 (M)
GPA: 4
WE:Analyst (Internet and New Media)
Schools: Cranfield (A) Warwick MiM "23 (M)
Posts: 3,095
dave13
Okay so if we start from start.
we know any no. ending with 1to9 leaves same remainder as its unit digit.
For ex- 11/10 leaves remainder 1. 29/10 leaves remainder 9 and so on.
so we just have to find what's the unit digit of 7^n+5, cool.
now 1st statement says when n is divided by 4 it leaves remainder 1.
then value of n = 1,5,9,13,17............
as stated above 7 unit digit repeats after every 4th power.
and 7^1 unit digit = 7^5 unit digit = 7^9 unit digit (the powers are at the gap of 4)
so the unit digit of every power of 7^5 in this statement is 7
and adding 5 in 7 will give you 12 or simply 2 as unit digit, which will be same when we divide the no. ending with 2 as unit digit with 10.
hence its sufficient.
Similarly for other statement.
Hopefully this helps.
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