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What is the tens digit of 6^17? [#permalink]
 03 Feb 2012, 20:44
Question Stats:
46% (02:07) correct
54% (01:21) wrong based on 7 sessions
What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9
Last edited by Bunuel on 04 Feb 2012, 05:37, edited 1 time in total.
Added the OA
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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.
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Re: What is the tens digit of 6^17? [#permalink]
10 Feb 2012, 03:14
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
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Re: What is the tens digit of 6^17? [#permalink]
10 Feb 2012, 05:10
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
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Re: What is the tens digit of 6^17? [#permalink]
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
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Re: What is the tens digit of 6^17? [#permalink]
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..
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Re: What is the tens digit of 6^17? [#permalink]
10 Feb 2012, 07:16
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Re: What is the tens digit of 6^17? [#permalink]
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 = 366^3 = 36 * 6 = n166^5 = 6^2 * 6^3 = 36 * n16 = n76Note 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 = n766^7 = 6^5 * 6^2 = n36 * n76 = n366^{17} = 6^{10} * 6^{7} = n76 * n36 (We already know what happens to n76 * n36 as calculated above...) =n36Answer: B
Last edited by mbaiseasy on 13 Jan 2013, 23:11, edited 4 times in total.
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Re: What is the tens digit of 6^17? [#permalink]
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 
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Re: What is the tens digit of 6^17? [#permalink]
14 Jan 2013, 01:44
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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
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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: math-number-theory-88376.htmlHope it helps.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory
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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. NEW!!!
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. NEW!!!
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