W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W. If W = X - 7, what is the sum of all possible values of W?

Question: W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W. If W = X - 7, what is the sum of all possible values of W?

Solution:
From the given information, we can write: X = YZ + W
Also given: W = X - 7

X = YZ + W
⇒ X = YZ + X - 7
⇒ 0 = YZ - 7
∴ 7 = YZ

There are only 2 possible cases:
case 1: Y = 1 and Z = 7
case 2: Y = 7 and Z = 1

case 1 yields a CONTRADICTION.
If Y = 1, then we are dividing X by 1, and if we divide by 1, the remainder will always be ZERO.
In other words, if Y = 1, then W = 0 So, we can definitely rule out case 1,

It must be the case that Y = 7 and Z = 1 (case 2)

So, we have: When X is divided by 7, the quotient is 1 and the remainder is W
This tells us that 7 divides into X 1 time
So, the possible values of X are: 7, 8, 9, 10, 11, 12 and 13 (since 7 divides into each value 1 time.

Let's check each case.
If X = 7,
then the remainder (W) is 0.
Doesn't follow the rule.

If X = 8,
then the remainder (W) is 1.
Since Y = W,
Doesn't follow the rule.

If X = 9,
then the remainder (W) is 2.
So, when X (9) is divided by 7 (Y), the quotient (Z) is 1, and the remainder (W) is 2.
Follows the rule.

If X = 10,
then the remainder (W) is 3.
So, when X (10) is divided by 7 (Y), the quotient (Z) is 1, and the remainder (W) is 3.
Follows the rule.

If X = 11,
then the remainder (W) is 4.
So, when X (11) is divided by 7 (Y), the quotient (Z) is 1, and the remainder (W) is 4.
Follows the rule.

If X = 12,
then the remainder (W) is 5.
So, when X (12) is divided by 7 (Y), the quotient (Z) is 1, and the remainder (W) is 5.
Follows the rule.

If X = 13,
then the remainder (W) is 6.
So, when X (13) is divided by 7 (Y), the quotient (Z) is 1, and the remainder (W) is 6.
Follows the rule.


Sum = 2 + 3 + 4 + 5 + 6
= 20