Problem:
Call a positive real number special if it has a decimal representation that consists entirely of digits 0 and 7. For example, 99700=7.07=7.070707… and 77.007 are special numbers. What is the smallest n such that 1 can be written as a sum of n special numbers?
Answer Choices:
A. 7
B. 8
C. 9
D. 10
E. 1 cannot be represented as a sum of finitely many special numbers
Solution:
Suppose I=x1+x2+⋯+xn where x1,x2,…,xn are special and n≤9. For k=1,2,3,…, let ak be the number of elements of {x1,x2,…,xn} whose kth decimal digit is 7. Then
1=107a1+1027a2+1037a3+⋯,
which yields
71=0.142857=10a1+102a2+103a3+⋯.
Hence a1=1,a2=4,a3=2,a4=8, ctc. In particular, this implies that n≥8. On the other hand,
x1=0.700,x2=x3=0.07,x4=x5=0.077777, and x6=x7=x8=0.000777 arc 8 special numbers whose sum is
999999700700+2(70707)+2(77777)+3(777)=1.
Thus the smatlest n is 8.