MathDB
Special Numbers

Source:

January 18, 2009

Problem Statement

Call a positive real number special if it has a decimal representation that consists entirely of digits 0 0 and 7 7. For example, \frac{700}{99} \equal{} 7.\overline{07} \equal{} 7.070707\cdots and 77.007 77.007 are special numbers. What is the smallest n n such that 1 1 can be written as a sum of n n special numbers? <spanclass=latexbold>(A)</span> 7<spanclass=latexbold>(B)</span> 8<spanclass=latexbold>(C)</span> 9<spanclass=latexbold>(D)</span> 10<spanclass=latexbold>(E)</span> The number 1 cannot be represented as a sum of finitely many special numbers. <span class='latex-bold'>(A)</span>\ 7\qquad <span class='latex-bold'>(B)</span>\ 8\qquad <span class='latex-bold'>(C)</span>\ 9\qquad <span class='latex-bold'>(D)</span>\ 10\qquad\\ <span class='latex-bold'>(E)</span>\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.}