The given question is a challenging GMAT 650 level quant problem solving question combining concepts in the Algebra and Number Properties.

Question 40: If x and y are non-negative integers such that 4x + 7y = 68, how many values are possible for (x + y)?

- A. 11
- B. 3
- C. 5
- D. 17
- E. 14

@ INR

**Given data**:

x and y are non-negative integers.

So, both x and y can take 0 or positive integer values.

4x + 7y = 68

=> 4x = 68 - 7y

Because x is an integer, 4x is divisible by 4. So, (68 - 7y) is divisible by 4.

68 is divisible by 4. So, **7y should also be divisible by 4**.

**x and y are non-negative integers.**

So, the least possible value for y is 0.

So, 7y = 0. __Note__: 0 is divisible by 4.

Subsequently, let us plug in multiples of 4 for y till such time x remains non-negative.

When y = 0, x = 17

When y = 4, x = 10

When y = 8, x = 3

When y = 12, x = -4 (x is negative)

For values of y that are multiples of 4 and are greater than 8, **x will be negative.**

**Possible values for x + y:**

17 + 0 = 17

10 + 4 = 14

8 + 3 = 11

**3 values are possible**

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