If $w x y z$ is a four-digit positive integer with $w \neq 0$, the layer sum of this integer equals $w x y z+x y z+y z+z$. For example, the layer sum of 4089 is $4089+089+89+9=4276$. If the layer sum of wxyz equals 2014, what is the value of $w+x+y+z$ ?
(A) 12
(B) 15
(C) 11
(D) 13
(E) 10

## Part C: Each correct answer is worth 8.
The layer sum of wxyz equals 2014.

This means that the sum of the integer with digits $w x y z$, the integer with digits $x y z$, the integer with digits $y z$, and the integer $z$ is 2014 .

Note that the integer with digits wxyz equals $1000 w+100 x+10 y+z$, the integer with digits $x y z$ equals $100 x+10 y+z$, and the integer with digits $y z$ equals $10 y+z$.

Therefore, we have

$$
(1000 w+100 x+10 y+z)+(100 x+10 y+z)+(10 y+z)+z=2014
$$

or

$$
1000 w+200 x+30 y+4 z=2014 \quad(*)
$$

Each of $w, x, y, z$ is a single digit and $w \neq 0$.

Now $w$ cannot be 3 or greater, or the left side of $(*)$ would be at least 3000, which is too large. Thus, $w=1$ or $w=2$.

If $w=2$, then $2000+200 x+30 y+4 z=2014$ and so $200 x+30 y+4 z=14$ or $100 x+15 y+2 z=7$. This would mean that $x=y=0$ (since otherwise the terms $100 x+15 y$ would contribute more than 7), which gives $2 z=7$ which has no integer solutions. Thus, $w \neq 2$.

Therefore, $w=1$.

This gives $1000+200 x+30 y+4 z=2014$ and so $200 x+30 y+4 z=1014$ or $100 x+15 y+2 z=507$.

Since $0 \leq y \leq 9$ and $0 \leq z \leq 9$, then $0 \leq 15 y+2 z \leq 15(9)+2(9)=153$.

Since $100 x$ is a multiple of 100 and $0 \leq 15 y+2 z \leq 153$, then $100 x=400$ or $100 x=500$ so $15 y+2 z=507-400=107$ or $15 y+2 z=507-500=7$. From above, we saw that $15 y+2 z$ cannot equal 7 , so $15 y+2 z=107$, which means that $100 x=400$ or $x=4$.

Thus, $15 y+2 z=107$.

Since $2 z$ is even, then $15 y$ must be odd to make $15 y+2 z$ odd.

The odd multiples of 15 less than 107 are 15,45,75,105.

Since $0 \leq 2 z \leq 18$, then we must have $15 y=105$ or $y=7$. This gives $2 z=2$ or $z=1$.

Therefore, the integer wxyz is 1471. (Checking, $1471+471+71+1=2014$.)

Finally, $w+x+y+z=1+4+7+1=13$.

ANSWER: (D)