Number Theory

 

A school orders $99$ textbooks, all for the same price.   

When the bill for the total order comes, the first and last   

digits are obscured. What are the missing digits?   
$_5801.3_   

 

Consider that the price per book is multiplied by 99.   

And, multiplying by 99 incorporates multiplying by 9.    

 

The digits of the product of a number multiplied by 9    

will always sum to a total that is evenly divisible by 9.    

 

5 + 8 + 0 + 1 + 3 = 17  so the sum of the two missing     

numbers will have to add 1 or 10 or 19, etc. to that 17.     

 

Using brute force, I reckoned that 15801.39 is evenly

divided by 99.  So the missing numbers are 1 and 9.    

And the cost of each textbook was 159.61.    

_.