Problem Solving

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?
A) 19
B) 20
C) 21
D) 22
E) 23

Source: www.gmatprepnow.com
Difficulty level: 650 - 700

Brent@GMATPrepNow wrote: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?
A) 19
B) 20
C) 21
D) 22
E) 23

Source: www.gmatprepnow.com
Difficulty level: 650 - 700

Given: When X is divided by Y, the quotient is Z and the remainder is W.
------ASIDE-------
There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
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So, from the given information, we can write: X = YZ + W

Also given: W = X - 7
Take X = YZ + W and replace X with X - 7 to get: X = YZ + X - 7
Subtract X from both sides: 0 = YZ - 7
Rewrite as: 7 = YZ

We're told that Y and Z are positive INTEGERS. So, there are only 2 possible cases:
case i: Y = 1 and Z = 7
case ii: Y = 7 and Z = 1

case i 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
However, we are told that W is a POSITIVE integer.
So, we can definitely rule out case i, which means it MUST be the case that Y = 7 and Z = 1 (case ii)

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. Since we're told W is POSITIVE, we can ELIMINATE this case.
If X = 8, then the remainder (W) is 1. Since Y already equals 1, and since we're told the numbers are DIFFERENT, we can ELIMINATE this case.
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. PERFECT!
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. PERFECT!
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. PERFECT!
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. PERFECT!
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. PERFECT!

What is the sum of all possible values of W?
Sum = 2 + 3 + 4 + 5 + 6
= 20

Answer: B

Cheers,
Brent