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Math Expert V
Joined: 02 Sep 2009
Posts: 59125
The Units Digits of Big Powers  [#permalink]

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3
6
THE UNITS DIGIT OF A POWER

How to Determine the Units (last) Digit of $$(xyz)^n$$:

1. Last digit of $$(xyz)^n$$ is the same as that of $$z^n$$;
2. Determine the cyclicity number $$c$$ of $$z$$;
3. Find the remainder $$r$$ when $$n$$ divided by the cyclisity;
4. When $$r>0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^r$$ and when $$r=0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^c$$, where $$c$$ is the cyclisity number.

• 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.
• Integers ending with 4 (eg. $$(xy4)^n$$) have a cyclisity of 2. When n is odd $$(xy4)^n$$ will end with 4 and when n is even $$(xy4)^n$$ will end with 6.
• Integers ending with 9 (eg. $$(xy9)^n$$) have a cyclisity of 2. When n is odd $$(xy9)^n$$ will end with 9 and when n is even $$(xy9)^n$$ will end with 1.

Example: What is the last digit of $$127^{39}$$?
Solution: Last digit of $$127^{39}$$ is the same as that of $$7^{39}$$. Now we should determine the cyclisity of $$7$$:

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of $$127^{39}$$ is the same as that of the last digit of $$7^{39}$$, is the same as that of the last digit of $$7^3$$, which is $$3$$.

Find below links to units digits, exponents, remainders, cyclicity problems.

Problem Solving:

650+

http://gmatclub.com/forum/what-is-the-t ... 27023.html
http://gmatclub.com/forum/if-you-divide ... 83350.html
http://gmatclub.com/forum/if-n-is-a-pos ... 05067.html
http://gmatclub.com/forum/what-is-the-u ... 01015.html
http://gmatclub.com/forum/if-n-is-a-pos ... 96262.html
http://gmatclub.com/forum/what-is-the-r ... 41050.html
http://gmatclub.com/forum/find-the-ones ... 41071.html
http://gmatclub.com/forum/m12-72970.html
http://gmatclub.com/forum/if-a-and-b-ar ... 87301.html

700+

http://gmatclub.com/forum/when-51-25-is ... 30220.html
http://gmatclub.com/forum/m25-73474.html
http://gmatclub.com/forum/which-of-the- ... 68179.html
http://gmatclub.com/forum/what-is-the-r ... 34778.html
http://gmatclub.com/forum/what-is-the-r ... 54889.html
http://gmatclub.com/forum/if-n-33-43-43 ... 40037.html

750+

http://gmatclub.com/forum/what-is-the-u ... 26681.html
http://gmatclub.com/forum/what-is-the-r ... 26493.html
http://gmatclub.com/forum/what-is-the-r ... 00316.html
http://gmatclub.com/forum/algebra-m26-145109.html
http://gmatclub.com/forum/what-is-the-r ... 56379.html
http://gmatclub.com/forum/what-is-the-r ... 99724.html

Data Sufficiency:

600+

http://gmatclub.com/forum/if-k-is-a-pos ... 26478.html
http://gmatclub.com/forum/if-r-s-and-t- ... 87298.html

650+

http://gmatclub.com/forum/if-x-is-a-pos ... 09075.html
http://gmatclub.com/forum/if-243-x-463- ... 02054.html
http://gmatclub.com/forum/if-r-s-and-t- ... 36746.html
http://gmatclub.com/forum/if-x-and-y-ar ... 87302.html

700+

http://gmatclub.com/forum/tough-and-tri ... l#p1029239
http://gmatclub.com/forum/if-x-and-y-ar ... 09636.html
_________________
Intern  B
Joined: 04 Mar 2017
Posts: 7
Re: The Units Digits of Big Powers  [#permalink]

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I have learnt all except 7, seems useful for GMAT Re: The Units Digits of Big Powers   [#permalink] 04 Feb 2019, 18:41
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