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Division reconstruction

Mr F. is a member of the University IT Service Unit and wants to find out how often and how long students of Faculty V are online. He sums up the durations of all their online sessions in the past year, this sum is  w  seconds. Then he divides  w  by the total number  y  of Faculty V students and gets  z , the average online time per student.  z  happens to be a natural number.

As F. deals with confidential data he writes down his calculation encoded. When he finishes this he accidentally spills some tea over the sheet. Only this remains of his calculation:



How many students has Faculty V ?

Solution

The number of digits in  w  and  z  can be concluded from the division steps.  y  has three digits because the multiples of  y  have three or four digits. These digit numbers and other immediately recognized figures are used to produce the following first solution step:



The second step is crucial for the determination of  y :



As  999 = 3 · 3 · 3 · 37 ,  y  must be  111, 333  or  999 .  111  can be ruled out because it cannot produce four-digit multiples in the division process.  999  can be ruled out because it cannot produce the difference  100  in the first division step.

Faculty V has  333  students.