## 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: $500$