#2           55. ALTO TERRIBLE!                  
The manager of the ‘Do-Re-Mi’ music shop frowned when he saw the week’s sale figures:

“Not very noteworthy,” he bached at his quavering accountant. “Find the individual sales figures of our staff.”
Music Department:  $1000
Woodwind Department:  $1200
Brass Department:    $1300
   $3500

The accountant soon gave him the liszt, and then beat a hasty retreat to the bar.
This was the score:

Bygrave:
Couchman
Davey
Foster
Greer 		$49
$91
$105
$129
$147		 Herni  
John
Kaehne  
Lillywhite  
Moore  		 $168
$189
$198
$210
$217		 Pope
Rawnsley  
Smith
Torry
Walters 		$303
$345
$357
$441
$551

“Well, that’s of minimal use,” muttered the manger, although the figures do add to $3500. He turned to his shop boy. “William, how can I Tell which names belong to which departments?”

Looking at the figures, William said, “Fortunately, 7 is my lucky number,” and he soon had the answer. Can you do as well?
 Hints and strategies 

Hint 1                       Hint 2                   Solution                      Extensions
HINT 1

This is a rather harder problem than some in this collection. Use the hint about 7: find the remainder of each number in the problem on division by 7. (If you know about congruence, find the residues of the numbers modulo 7.)

HINT 2

You should have obtained remainders 6, 3, 7 for the respective departments. Also remainders 3, 2, 2, 2, 5 for F (Foster), K, P, R and W; all other names have remainder 0. You should now immediately be able to place five people in their departments.
SOLUTION

As in Hint 2, we obtained remainders 6, 3, 7 for the respective departments. Also remainders 3, 2, 2, 2, 5 for F (Foster), K, P, R and W; all other names have remainder 0. Since the sum of the remainders for the summands (people) must be the same as the remainders for the departments they belong to, we can now immediately able place five people in their departments:

K, P and R belong to the Music Department, F belongs to the Woodwind Department, and W belongs to the Brass Department.

To place the others, we must use a little trial and error. I suggest starting with the larger people amounts. We obtain:

Music: 198 + 303 + 345 + 49 + 105 (K, P, R, B and D);
Woodwind: 129 + 147 + 168 + 189 + 210 + 357 (F, G, H, J, L and S);
Brass: 551 + 91 + 217 + 441 (W, C, M and T).
EXTENSION

1. Try constructing a problem of your own. You will need a set of numbers broken up into summands (smaller parts), and some number (not necessarily 7) which divides a lot of them. And some story to make the problem more interesting (you can leave out the dreadful puns!).

2. Does the given problem work with some number other than 7? (Probably not.) The strategy used here depends on many of the numbers having the same remainder when divided by 7 (or some other divisor).