Difference between revisions of "2018 AMC 10A Problems/Problem 18"
Ishankhare (talk | contribs) (Created page with "How many nonnegative integers can be written in the form <cmath>a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,</cmath> where...") |
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\textbf{(D) } 3281 \qquad | \textbf{(D) } 3281 \qquad | ||
\textbf{(E) } 59,048 </math> | \textbf{(E) } 59,048 </math> | ||
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+ | ==Solution== | ||
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+ | This looks like balanced ternary, in which all the integers with absolute values less than <math>\frac{3^n}{2}</math> are represented in <math>n</math> digits. There are 8 digits. Plugging in 8 into the formula gives a maximum bound of <math>|x|=3280.5</math>, which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are <math>3280+1=\boxed{3281}</math> integers or <math>\boxed{D}</math>. |
Revision as of 18:08, 8 February 2018
How many nonnegative integers can be written in the form
where
for
?
Solution
This looks like balanced ternary, in which all the integers with absolute values less than are represented in
digits. There are 8 digits. Plugging in 8 into the formula gives a maximum bound of
, which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are
integers or
.