2018 AMC 12A Problem #13

Number Theory Level 2

How many nonnegative integers can be written in the form $a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0,$ where $a_i \in \{-1,0,1\}$ for $0 \leq i \leq 7$ ?

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Ryan Ong
Feb 9, 2018

For every representation of $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0$ . that equals a positive number $x$ . There is always a representation that equals $-x$ . Just times our initial representation of $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0$ that equals positive number x by -1. Thus, #(positive representations) = #(negative representations). Since each $a_i$ can have 3 choices of numbers.The answer is $((3^8-1)/2) + 1$

We subtracted 1 because 0 is neither a positive or negative number.

Jerry McKenzie - 3 years, 4 months ago

2018 AMC 12A Problem #13

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