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...") |
|||
Line 7: | Line 7: | ||
\textbf{(D) } 3281 \qquad | \textbf{(D) } 3281 \qquad | ||
\textbf{(E) } 59,048 </math> | \textbf{(E) } 59,048 </math> | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | 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 17:08, 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 .