GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 11 Dec 2019, 14:30

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

S97-15

Author Message
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 59674

Show Tags

16 Sep 2014, 01:51
00:00

Difficulty:

65% (hard)

Question Stats:

54% (01:59) correct 46% (01:35) wrong based on 61 sessions

HideShow timer Statistics

The tens digit of $$6^{17}$$ is

A. 1
B. 3
C. 5
D. 7
E. 9

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 59674

Show Tags

16 Sep 2014, 01:51
Official Solution:

The tens digit of $$6^{17}$$ is

A. 1
B. 3
C. 5
D. 7
E. 9

We know that there must be a pattern, since we can’t be expected to expand $$6^{17}$$ out to all its digits. In other words, we must be able to spot a repeating cycle of digits.

The only way forward is to compute tens digits for powers of 6, starting with $$6^1$$, and see what we get. To go up, multiply the previous result by 6 and drop any higher digits than the tens, but we have to keep the units digit (which, as we’ll see, will be 6 every time).

$$6^1 = 6$$ (no tens digit)

$$6^2 = 6 \times 6^1 = 36$$ (tens digit = 3)

$$6^3 = 6 \times 6^2 = ..16$$ (tens digit = 1)

$$6^4 = 6 \times 6^3 = ..96$$ (tens digit = 9)

$$6^5 = 6 \times 6^4 = ..76$$ (tens digit = 7)

$$6^6 = 6 \times 6^5 = ..56$$ (tens digit = 5)

$$6^7 = 6 \times 6^6 = ..36$$ (tens digit = 3)

Whew - the numbers finally started repeating! The cycle is 3, 1, 9, 7, 5 - which is 5 terms long. Every power will have the same tens digit as the 5th larger power, so $$6^2$$, $$6^7$$, $$6^{12}$$, and most importantly $$6^{17}$$ will all have 3 as their tens digit.

Notice that the pattern didn’t start until $$6^2$$. $$6^1$$ doesn’t have a tens digit (or has a tens digit of 0, but this digit is never repeated later in the cycle).

_________________
Intern
Joined: 17 Jan 2015
Posts: 13

Show Tags

17 Mar 2015, 07:27
It can also be done fast by breaking 6 into: 2 to the power 17 and 3 to the power 17.
and then check the cyclicity....

Intern
Joined: 07 Nov 2016
Posts: 1

Show Tags

07 Dec 2017, 10:11
Math Expert
Joined: 02 Sep 2009
Posts: 59674

Show Tags

07 Dec 2017, 10:14
1
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.

_________________
Intern
Joined: 24 Jul 2014
Posts: 2

Show Tags

18 Sep 2018, 11:51
any alternate solution
Math Expert
Joined: 02 Sep 2009
Posts: 59674

Show Tags

18 Sep 2018, 20:16
shivamlohiya007 wrote:
any alternate solution

Have you checked post just above yours?

https://gmatclub.com/forum/s97-184703.html#p1974800
_________________
Re: S97-15   [#permalink] 18 Sep 2018, 20:16
Display posts from previous: Sort by

S97-15

Moderators: chetan2u, Bunuel