, 2019

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: [[IMAGE0]] Finally, $P+Q+R+S+T=1+0+2+5+6=14$.

Solution 2

Let $x$ be the five-digit integer with digits $PQRST$.

This means that $PQRST0=10x$ and so $PQRST4=10x+4$.

Also, $4PQRST=400\,000+PQRST = 400\,0000 + x$.

From the given multiplication, $4(10x+4)=400\,000+x$ which gives $40x + 16 = 400\,000+x$ or $39x = 399\,984$.

Thus, $x = \dfrac{399\,984}{39} = 10\,256$.

Since $PQRST=10\,256$, then $P+Q+R+S+T=1+0+2+5+6=14$.