Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 29 Nov 2011
Posts: 80

What is the tens digit of 6^17? [#permalink]
Show Tags
03 Feb 2012, 20:44
24
This post was BOOKMARKED
Question Stats:
48% (02:19) correct
52% (01:24) wrong based on 832 sessions
HideShow timer Statistics
What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9
Official Answer and Stats are available only to registered users. Register/ Login.
Last edited by Bunuel on 09 Mar 2014, 11:55, edited 2 times in total.
Added the OA



Math Expert
Joined: 02 Sep 2009
Posts: 39713

Re: Arithmetic [#permalink]
Show Tags
04 Feb 2012, 05:30
5
This post received KUDOS
Expert's post
13
This post was BOOKMARKED
Smita04 wrote: What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition. The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits). Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3. Answer: B.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Director
Joined: 23 Apr 2010
Posts: 581

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
10 Feb 2012, 03:14
2
This post received KUDOS
2
This post was BOOKMARKED
Bunuel,can you calculate it with modulo as below:
1) periodicity of the ten's digit is 5 2) 17 mod 5 = 2 3) 6^17 will have the same digit as 6^2



Director
Status: Enjoying the GMAT journey....
Joined: 26 Aug 2011
Posts: 718
Location: India

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
10 Feb 2012, 05:10
1
This post received KUDOS
1
This post was BOOKMARKED
well, this question demands calculation to see a pattern of tens digits keep calculating till it's confirmed that u have hit a pattern. 6^1 = 6 6^2 = 36 6^3 = 216 now don't multiply 216 by 6, rather we are interested in only first two digits to know the outcome so 6^4 = 96 ( 16 x 6) 6^5 = 576 ( 96 x 6) 6^ 6 = 456 ( 76 x 6) 7^ 6 =336 ( 56 x 6) so now we have the pattern in tens digit i.e. 3 in (6^2), 1 in (6^3), 9 in (6^4), 7 in (6^5), 5 in (6^6), 3 in (6^7), so the tens digit is 3 for the 2,7,12 and 17 times.. IMO B
_________________
Fire the final bullet only when you are constantly hitting the Bull's eye, till then KEEP PRACTICING.
A WAY TO INCREASE FROM QUANT 3540 TO 47 : http://gmatclub.com/forum/awaytoincreasefromq3540toq138750.html
Q 47/48 To Q 50 + http://gmatclub.com/forum/thefinalclimbquestforq50fromq47129441.html#p1064367
Three good RC strategies http://gmatclub.com/forum/threedifferentstrategiesforattackingrc127287.html



Director
Joined: 23 Apr 2010
Posts: 581

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
10 Feb 2012, 05:23
I know how to find the cyclicity of the ten's digit. I just wanted to know whether steps 2) and 3) can be used: Quote: 2) 17 mod 5 = 2 3) 6^17 will have the same digit as 6^2



Manager
Joined: 10 Jan 2010
Posts: 185
Location: Germany
Concentration: Strategy, General Management
GPA: 3
WE: Consulting (Telecommunications)

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
10 Feb 2012, 06:15
1. Identify pattern (first one does not have a tens, 3, 1, 9, 7, 5, 3) 2. scale up to 6^17 = tens digit 3
Pick B as the answer..



Math Expert
Joined: 02 Sep 2009
Posts: 39713

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
10 Feb 2012, 07:16



Senior Manager
Joined: 13 Aug 2012
Posts: 464
Concentration: Marketing, Finance
GPA: 3.23

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
21 Dec 2012, 07:38
Smita04 wrote: What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 \(6^2 = 36\) \(6^3 = 36 * 6 = n16\) \(6^5 = 6^2 * 6^3 = 36 * n16 = n76\) Note that when you multiply, you don't have to finish it all the way, knowing the tens digit should suffice.... Also, using the table we have we can calculate \(6^{10}\) and \(6^{17}\). We work with what we already have above/\(6^10 = 6^5 * 6^5 = n76 * n76 = n76\) \(6^7 = 6^5 * 6^2 = n36 * n76 = n36\) \(6^{17} = 6^{10} * 6^{7} = n76 * n36\) (We already know what happens to n76 * n36 as calculated above...) \(=n36\) Answer: B
_________________
Impossible is nothing to God.
Last edited by mbaiseasy on 13 Jan 2013, 23:11, edited 4 times in total.



Manager
Joined: 04 Jan 2013
Posts: 80

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
13 Jan 2013, 13:15
bunuel would you please post me a link on the topic of exponents and powers from gmat math book if it has been finished..i want to learn and master way u hav solved the problem
Posted from my mobile device



Math Expert
Joined: 02 Sep 2009
Posts: 39713

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
14 Jan 2013, 01:44



Manager
Joined: 04 Jan 2013
Posts: 80

Re: Arithmetic [#permalink]
Show Tags
17 Jan 2013, 10:59
Bunuel wrote: Smita04 wrote: What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition. The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits). Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3. Answer: B. i have noticed that every number has 6 as the unit digit..is it the same for other numbers that they repeat each of the unit's digit throughout when it is being raised to powers of consecutive integers Posted from my mobile device



Math Expert
Joined: 02 Sep 2009
Posts: 39713

Re: Arithmetic [#permalink]
Show Tags
18 Jan 2013, 04:28
chiccufrazer1 wrote: Bunuel wrote: Smita04 wrote: What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition. The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits). Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3. Answer: B. i have noticed that every number has 6 as the unit digit..is it the same for other numbers that they repeat each of the unit's digit throughout when it is being raised to powers of consecutive integers Posted from my mobile device No. You could test that very easily yourself. Is the units digit of 2^2 equal 2? No, its 4. • Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. • Integers ending with 2, 3, 7 and 8 have a cyclicity of 4. For more check here: http://gmatclub.com/forum/mathnumbertheory88376.htmlHope it helps.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Math Expert
Joined: 02 Sep 2009
Posts: 39713

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
09 Mar 2014, 13:08



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15995

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
04 May 2015, 20:22
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources



Manager
Joined: 10 Mar 2014
Posts: 245

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
02 Jul 2015, 10:56
Bunuel wrote: Smita04 wrote: What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition. The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits). Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3. Answer: B. . Hi Bunuel, Can we do this in following way 6^17 = (2*3)^17 (2^16*3^16) *2*3 now 2 repeats in pattern 2,4,8,6 and 3 repeats in pattern 3,9,7,1 so when we multiply 2^16*3^16 last digit is 6 now multiply 6 *6 so its 36 so tens digit is 3. Please clarify Thanks



Manager
Joined: 09 Jun 2015
Posts: 101

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
20 Mar 2016, 02:05
Smita04 wrote: What is the tens digit of 6^17?
(A) 1 (B) 3 (C) 5 (D) 7 (E) 9 6^4 when divided by 100 the remainder is 4 So we have to check 4^4*6 when divided by 100 or 256*6 = 1536



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15995

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
20 Apr 2017, 23:02
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources



Intern
Joined: 06 Oct 2016
Posts: 4
Location: Germany

Re: What is the tens digit of 6^17? [#permalink]
Show Tags
21 Apr 2017, 02:22
6^17 = 16,926,659,444,736 & its tenth digit is 3.
option b is correct.




Re: What is the tens digit of 6^17?
[#permalink]
21 Apr 2017, 02:22







