THE UNITS DIGIT OF A POWERHow 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.htmlhttp://gmatclub.com/forum/if-you-divide ... 83350.htmlhttp://gmatclub.com/forum/if-n-is-a-pos ... 05067.htmlhttp://gmatclub.com/forum/what-is-the-u ... 01015.htmlhttp://gmatclub.com/forum/if-n-is-a-pos ... 96262.htmlhttp://gmatclub.com/forum/what-is-the-r ... 41050.htmlhttp://gmatclub.com/forum/find-the-ones ... 41071.htmlhttp://gmatclub.com/forum/m12-72970.htmlhttp://gmatclub.com/forum/if-a-and-b-ar ... 87301.html700+

http://gmatclub.com/forum/when-51-25-is ... 30220.htmlhttp://gmatclub.com/forum/m25-73474.htmlhttp://gmatclub.com/forum/which-of-the- ... 68179.htmlhttp://gmatclub.com/forum/what-is-the-r ... 34778.htmlhttp://gmatclub.com/forum/what-is-the-r ... 54889.htmlhttp://gmatclub.com/forum/if-n-33-43-43 ... 40037.html750+

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

600+

http://gmatclub.com/forum/if-k-is-a-pos ... 26478.htmlhttp://gmatclub.com/forum/if-r-s-and-t- ... 87298.html650+

http://gmatclub.com/forum/if-x-is-a-pos ... 09075.htmlhttp://gmatclub.com/forum/if-243-x-463- ... 02054.htmlhttp://gmatclub.com/forum/if-r-s-and-t- ... 36746.htmlhttp://gmatclub.com/forum/if-x-and-y-ar ... 87302.html700+

http://gmatclub.com/forum/tough-and-tri ... l#p1029239http://gmatclub.com/forum/if-x-and-y-ar ... 09636.html
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