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# Units Digit

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Intern
Joined: 17 Apr 2010
Posts: 38

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22 Apr 2010, 07:16
Hi everybody!

I got a question concerning the pattern of getting the units digit in questions like the following:

What is the units digit of 3^17 × 17^54 × 135^19 × 31^21 × 42^17?

Is it true that the units digit is repeated with every 4th power, i.e. 3^2 has the same units digit as 3^6 and 3^10?

My approach:

3^17 ---> 17 divided by 4, remainder 1---> 3^1 ---> Units digit 3
17^54 ---> 54 divided by 4, remainder 2---> 7^2 ---> Units digit 9
135^19 ---> 19 divided by 4, remainder 3---> 5^3 ---> Units digit 5
31^21 ---> 21 divided by 4, remainder 1---> 1^1 ---> Units digit 1
42^17 ---> 17 divided by 4, remainder 1---> 2^1 ---> Units digit 2

3 x 9 x 5 x 1 x 2 = 270

Would this be a good approach?
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22 Apr 2010, 07:24
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I think you are missing one point here.

The pattern of 4 is not for every number.

Check this link out for cyclicity.

http://takshzilabeta.com/index.php?opti ... -&Itemid=5

In order to solve this very problem, I think you should go through the following thread.

compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html

Kindly do share the answer of this question too.
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CEO
Joined: 17 Nov 2007
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Schools: Chicago (Booth) - Class of 2011
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22 Apr 2010, 07:40
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Expert's post
1,1,1,1,1
2,4,8,6,2
3,9,7,1,3
4,6,4,6,4
5,5,5,5,5
6,6,6,6,6
7,9,3,1,7
8,4,2,6,8
9,1,9,1,9
0,0,0,0,0

You can also use that:
1,5,6,0 is repeated with every 1st power (You can ignore power)
4,9 is repeated with every 2nd power (odd|even power = 1st|2nd power)
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Intern
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22 Apr 2010, 09:11
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Manager
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23 Apr 2010, 05:13
As I understand it, whenever there is a product of multiple of 5's and multiple of 2's , we will always end up with a last digit of zero's. Example:
5 ^1* 2^1 => last digit is 0
5^2 * 2^2 => last 2 digits are 0
so 5^19 * 2^17 will have 17 0's on the right hand side. Someone please correct me if I'm wrong...

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23 Apr 2010, 05:20
Eden, you are right.
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14 Jan 2011, 09:50
Pls. find everybody very useful the reference of hussain15 to the link
http://takshzilabeta.com/index.php?opti ... -&Itemid=5

Thank you very much Hussain!!!

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Intern
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15 Mar 2012, 12:55
Hi all,
i was just wondering if it is always possible to use the approach of "intern" with those kind of questions?

It seems the most easiest way to me.

Many thanks

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16 Mar 2012, 04:06
NYC5648 wrote:
Hi all,
i was just wondering if it is always possible to use the approach of "intern" with those kind of questions?

It seems the most easiest way to me.

Many thanks

Check this: math-number-theory-88376.html. Chapters: LAST DIGIT OF A PRODUCT and LAST DIGIT OF A POWER are dedicated to this subject.
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Re: Units Digit   [#permalink] 16 Mar 2012, 04:06
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