In the multiplication shown, each of $P, Q, R, S$, and $T$ is a digit. The value of $P+Q+R+S+T$ is
(A) 14
(B) 20
(C) 16
(D) 17
(E) 13

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-028.jpg?height=146&width=379&top_left_y=2152&top_left_x=1361)
Solution 1

We start with the ones digits.

Since $4 \times 4=16$, then $T=6$ and we carry 1 to the tens column.

Looking at the tens column, since $4 \times 6+1=25$, then $S=5$ and we carry 2 to the hundreds column.

Looking at the hundreds column, since $4 \times 5+2=22$, then $R=2$ and we carry 2 to the thousands column.

Looking at the thousands column, since $4 \times 2+2=10$, then $Q=0$ and we carry 1 to the ten thousands column.

Looking at the ten thousands column, since $4 \times 0+1=1$, then $P=1$ and we carry 0 to the hundred thousands column.

Looking at the hundred thousands column, $4 \times 1+0=4$, as expected.

This gives the following completed multiplication:

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-058.jpg?height=162&width=331&top_left_y=838&top_left_x=954)

Finally, $P+Q+R+S+T=1+0+2+5+6=14$.

## Solution 2

Let $x$ be the five-digit integer with digits $P Q R S T$.

This means that $P Q R S T 0=10 x$ and so $P Q R S T 4=10 x+4$.

Also, $4 P Q R S T=400000+P Q R S T=4000000+x$.

From the given multiplication, $4(10 x+4)=400000+x$ which gives $40 x+16=400000+x$ or $39 x=399984$.

Thus, $x=\frac{399984}{39}=10256$.

Since $P Q R S T=10256$, then $P+Q+R+S+T=1+0+2+5+6=14$.

ANSWER: (A)