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.