Units Digit of Seven

\left \{3^0,3^1,3^2,3^3,\cdots,3^{2000}\right \}
How many numbers in the above set will have a ones digit of seven?
Source: mathcontest.olemiss.edu 4/28/20134

SOLUTION
3^3=27
3^7=81\cdot 27=2187
3^{11}=81^2\cdot 27=177147
3^{15}=81^3\cdot 27=14348907
3^{19}=81^4\cdot 27=1162261467
\cdots
3^{1995}=81^{498}\cdot 27=xxx7
3^{1999}=81^{499}\cdot 27=xxx7
Imagine the exponents 3,4,5,6,\cdots,1998,1999,2000 plotted on the number line. Of these exponents we want 3,7,11,15,\cdots,1995,1999.
How many of them?
\dfrac{2000-3}{4}=499
Numbers in the set that will have a ones digit of seven
499+1=500

Answer: 500

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , . Bookmark the permalink.

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