The weak statement in your logic reasoning is following:

**Quote:**

**It seems that the minimum amount that each of the remaining 8 winners could receive is $360/8 = $45.**

$360/8 = $45 is the AVERAGE prize that one of the 8 remaining winners gets, NOT the minimal one. The minimal prize is determined in question statement and it is $20.

It is possible that EACH one of the 8 remaining winners gets the same amount of money however in this case greatest award is just $45 but we've shown in explanation that greatest award can be as big as $220.

Each one gets the same:

$45 $45 $45 $45 $45 $45 $45 $45

7 pers. get minimal, 8th gets the rest:

$220 $20 $20 $20 $20 $20 $20 $20

Let us take a look at this situation once again. We have $360 to be distributed among 8 people, each one of whom must receive at least $20.

Can one person get all $360? No, because other 7 must get at least $20 or 7 × $20 = $140 overall. So the most money one person can get is $360 – $140 = $220. And it can not be any larger amount of money because if we give him just $1 more we'll have to take that $1 out of somebody's MINIMAL prize.