Z9-I-4

Kate thought a five-digit integer. She wrote the sum of this number and its half at the first line to the workbook. On the second line wrote a total of this number and its one fifth. On the third row, she wrote a sum of this number and its one nines. Finally, all three lines sum and result wrote on the fourth line. Then she was amazing found that on the third line has a written cube of a certain natural number.

Determine the smallest number Kate can think in the beginning.

Correct result:

n =  11250

Solution:

$9999 < n < 100000 \ \\ n=11250 \ \\ l_{1}=n+n/2=11250+11250/2=16875 \ \\ l_{2}=n+n/5=11250+11250/5=13500 \ \\ l_{3}=n+n/9=11250+11250/9=12500 \ \\ l_{4}=l_{1}+l_{2}+l_{3}=16875+13500+12500=42875 \ \\ l_{5}=35^3=42875 \ \\ n=11250$

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