In the multiplication shown, $P$ and $Q$ each represent a single digit, and the product is 32951 . What is the value of $P+Q$ ?
(A) 14
(B) 12
(C) 15
(D) 13
(E) 11

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-105.jpg?height=90&width=355&top_left_y=1798&top_left_x=1313)
Solution 1

Since the units digit of the product $39 P \times Q 3$ comes from multiplying $P \times 3$, and this units digit is a 1 , then $P$ must be the digit 7 .

Therefore, $397 \times Q 3=32951$ so $Q 3=\frac{32951}{397}=83$, so $Q=8$.

Thus, $P+Q=15$.

## Solution 2

Since the units digit of the product $39 P \times Q 3$ comes from multiplying $P \times 3$, and this units digit is a 1, then $P$ must be the digit 7 .

Therefore, the product becomes

| 397 |
| ---: |
| $\times \quad Q 3$ |
| 1191 |
| $\square$ |
| 32951 |

Therefore, the product $Q \times 7$ must have a units digit of 6 (where the $\square$ is) in order to get the tens digit of 5 in the product. Since $Q \times 7$ has a units digit of 6 , then $Q=8$. Thus, $P+Q=15$.

ANswer: (C)