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What is the tens digit of 6^17?

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What is the tens digit of 6^17? [#permalink] New post 03 Feb 2012, 19:44
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What is the tens digit of 6^17?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Mar 2014, 10:55, edited 2 times in total.
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Re: Arithmetic [#permalink] New post 04 Feb 2012, 04:30
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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] New post 10 Feb 2012, 02:14
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] New post 10 Feb 2012, 04:10
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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] New post 10 Feb 2012, 04:23
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] New post 10 Feb 2012, 05:15
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] New post 10 Feb 2012, 06:16
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Re: What is the tens digit of 6^17? [#permalink] New post 21 Dec 2012, 06:38
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

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Last edited by mbaiseasy on 13 Jan 2013, 22:11, edited 4 times in total.
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Re: What is the tens digit of 6^17? [#permalink] New post 13 Jan 2013, 12:15
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

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Re: What is the tens digit of 6^17? [#permalink] New post 14 Jan 2013, 00:44
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chiccufrazer1 wrote:
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

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For more on number theory and exponents check: math-number-theory-88376.html

DS questions on exponents: search.php?search_id=tag&tag_id=39
PS questions on exponents: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope it helps.

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Re: Arithmetic [#permalink] New post 17 Jan 2013, 09:59
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

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Re: Arithmetic [#permalink] New post 18 Jan 2013, 03:28
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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

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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.html

Hope it helps.

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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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.


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Re: What is the tens digit of 6^17? [#permalink] New post 09 Mar 2014, 12:08
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Re: What is the tens digit of 6^17?   [#permalink] 09 Mar 2014, 12:08
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