January 17, 2019 January 17, 2019 08:00 AM PST 09:00 AM PST Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL. January 19, 2019 January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 14 Feb 2017
Posts: 122
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23
GPA: 2.61
WE: Management Consulting (Consulting)

If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
Updated on: 10 Dec 2018, 20:36
Question Stats:
79% (01:10) correct 21% (01:41) wrong based on 82 sessions
HideShow timer Statistics
If K is a positive integer, what is the remainder when \(13^{4K+2} +8\) is divided by 10 ? A) 7 B) 4 C) 2 D) 1 E) 0
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by dcummins on 10 Dec 2018, 19:03.
Last edited by Bunuel on 10 Dec 2018, 20:36, edited 2 times in total.
Edited the question.




Math Expert
Joined: 02 Aug 2009
Posts: 7199

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
11 Dec 2018, 18:39
hibobotamuss wrote: I don't quite get this. Can someone explain a little better? Bunuel chetan2uHi... When you divide any number by 10, the remainder is the units digit.. WHY?.. Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder. So we have to find the units digit of the term. All digits have a cylicity when it comes to units digit... 3 gives 3,9,7,1,3,9,7,1,3,9..... \(3^1=3;3^2=9;3^3=27,3^4=81;3^5=243\)..unit digits are 3,9,7,1,3... Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity... Therefore \(13^{4k+2}\) will have same units digit as 13^2, so units digit will be 9.. So the units digit of entire term will be 9+8=17, thus 7 will be the remainder. A
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor




Director
Joined: 18 Jul 2018
Posts: 554
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
10 Dec 2018, 19:17
Bunuel sir, Can you rectify the question? is it \((13^{4K}+2)+8\) or \((13^4*K+2)+8\)
_________________
If you are not badly hurt, you don't learn. If you don't learn, you don't grow. If you don't grow, you don't live. If you don't live, you don't know your worth. If you don't know your worth, then what's the point?



Intern
Joined: 20 Aug 2017
Posts: 26

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
10 Dec 2018, 19:47
How the remainder is 7?



Director
Joined: 18 Jul 2018
Posts: 554
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
10 Dec 2018, 19:50
sghoshgt wrote: How the remainder is 7? sghoshgt, exactly, shouldn't the remainder be 1? Posted from my mobile device
_________________
If you are not badly hurt, you don't learn. If you don't learn, you don't grow. If you don't grow, you don't live. If you don't live, you don't know your worth. If you don't know your worth, then what's the point?



Manager
Joined: 14 Feb 2017
Posts: 122
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23
GPA: 2.61
WE: Management Consulting (Consulting)

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
Updated on: 10 Dec 2018, 20:00
To solve this problem examine the functioning of the units digit of \(13^4^k+2\)
Any integer >10 we can refer to the units digit to identify the pattern for subsequent powers of that digit as follows: \(13^1= 13\)
\(13^2 = 169\)
\(13^3 = 2197\)
\(13^4 = 28,561\)
This is the 3s units digit pattern \(3971\)
It has a cyclicity of 4, meaning it repeats each multiple of 4.
Thus units digit of 397139 add 8 = 17 1[/m]7 > Units digit of 7
Originally posted by dcummins on 10 Dec 2018, 19:55.
Last edited by dcummins on 10 Dec 2018, 20:00, edited 2 times in total.



Manager
Joined: 14 Feb 2017
Posts: 122
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23
GPA: 2.61
WE: Management Consulting (Consulting)

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
10 Dec 2018, 19:57
Afc0892 wrote: Bunuel sir, Can you rectify the question? is it \((13^{4K}+2)+8\) or \((13^4*K+2)+8\) I've rectified it. Apologies, I'm getting use to the forum code.



Director
Joined: 18 Jul 2018
Posts: 554
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
10 Dec 2018, 20:09
dcummins wrote: To solve this problem examine the functioning of the units digit of \(13^4^k+2\)
Any integer >10 we can refer to the units digit to identify the pattern for subsequent powers of that digit as follows: \(13^1= 13\)
\(13^2 = 169\)
\(13^3 = 2197\)
\(13^4 = 28,561\)
This is the 3s units digit pattern \(3971\)
It has a cyclicity of 4, meaning it repeats each multiple of 4.
Thus units digit of 397139 add 8 = 17 1[/m]7 > Units digit of 7 The question is 13 Whole to the power 4k+2 and not just \(13^{4k} + 2\)?
_________________
If you are not badly hurt, you don't learn. If you don't learn, you don't grow. If you don't grow, you don't live. If you don't live, you don't know your worth. If you don't know your worth, then what's the point?



Manager
Joined: 14 Feb 2017
Posts: 122
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23
GPA: 2.61
WE: Management Consulting (Consulting)

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
10 Dec 2018, 20:59
Afc0892 wrote: dcummins wrote: To solve this problem examine the functioning of the units digit of \(13^4^k+2\)
Any integer >10 we can refer to the units digit to identify the pattern for subsequent powers of that digit as follows: \(13^1= 13\)
\(13^2 = 169\)
\(13^3 = 2197\)
\(13^4 = 28,561\)
This is the 3s units digit pattern \(3971\)
It has a cyclicity of 4, meaning it repeats each multiple of 4.
Thus units digit of 397139 add 8 = 17 1[/m]7 > Units digit of 7 The question is 13 Whole to the power 4k+2 and not just \(13^{4k} + 2\)? I clarified this in the stem. Reread it please



Manager
Joined: 28 Jun 2018
Posts: 72

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
11 Dec 2018, 07:09
I don't quite get this. Can someone explain a little better? Bunuel chetan2u



Intern
Joined: 12 Sep 2017
Posts: 43

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
26 Dec 2018, 14:29
Hi... When you divide any number by 10, the remainder is the units digit.. WHY?.. Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.
So we have to find the units digit of the term. All digits have a cylicity when it comes to units digit... 3 gives 3,9,7,1,3,9,7,1,3,9..... \(3^1=3;3^2=9;3^3=27,3^4=81;3^5=243\)..unit digits are 3,9,7,1,3... Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity... Therefore \(13^{4k+2}\) will have same units digit as 13^2, so units digit will be 9.. So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.
A[/quote]
Thank you for your answer!
I just have one question, how do we know that the cyclicity number that we have to choose is 9?
Kind regards!



Math Expert
Joined: 02 Aug 2009
Posts: 7199

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
26 Dec 2018, 18:18
jfranciscocuencag wrote: Hi... When you divide any number by 10, the remainder is the units digit.. WHY?.. Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.
So we have to find the units digit of the term. All digits have a cylicity when it comes to units digit... 3 gives 3,9,7,1,3,9,7,1,3,9..... \(3^1=3;3^2=9;3^3=27,3^4=81;3^5=243\)..unit digits are 3,9,7,1,3... Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity... Therefore \(13^{4k+2}\) will have same units digit as 13^2, so units digit will be 9.. So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.
A Thank you for your answer! I just have one question, how do we know that the cyclicity number that we have to choose is 9? Kind regards![/quote] Hi .. We know cylicity is 3,9,7,1,3,9,... So the cylicity is in 4s...(3,9,7,1),(3,9,7,1).... So the cylicity is same for first, fifth, ninth terms.... And for second, sixth, tenth and so on. That is every fourth term is same... Here we have 13^{4k+2}... Now 4k means every fourth term, that is 13^{4k} will have _,_,_,1,_,_,_,1...that is it will have 1.. But we are looking for (4k+2)th term and it will have same number as the 2nd term, and, therefore, it is 9...
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



Intern
Joined: 12 Sep 2017
Posts: 43

Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is
[#permalink]
Show Tags
26 Dec 2018, 18:26
chetan2u wrote: jfranciscocuencag wrote: Hi... When you divide any number by 10, the remainder is the units digit.. WHY?.. Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.
So we have to find the units digit of the term. All digits have a cylicity when it comes to units digit... 3 gives 3,9,7,1,3,9,7,1,3,9..... \(3^1=3;3^2=9;3^3=27,3^4=81;3^5=243\)..unit digits are 3,9,7,1,3... Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity... Therefore \(13^{4k+2}\) will have same units digit as 13^2, so units digit will be 9.. So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.
A Thank you for your answer! I just have one question, how do we know that the cyclicity number that we have to choose is 9? Kind regards! Hi .. We know cylicity is 3,9,7,1,3,9,... So the cylicity is in 4s...(3,9,7,1),(3,9,7,1).... So the cylicity is same for first, fifth, ninth terms.... And for second, sixth, tenth and so on. That is every fourth term is same... Here we have 13^{4k+2}... Now 4k means every fourth term, that is 13^{4k} will have _,_,_,1,_,_,_,1...that is it will have 1.. But we are looking for (4k+2)th term and it will have same number as the 2nd term, and, therefore, it is 9...[/quote] +Kudos Thank you very much!!! So clear now.




Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is &nbs
[#permalink]
26 Dec 2018, 18:26






