No Guessing.

Number Theory Level 2

$\begin{array} { l l l l l } & & P & Q & R & S& T& 4 \\\times &&&&&&&4\\\hline & \\ &&\ 4 & P & Q& R & S &T \end{array}$

What is $P+Q+R+S+T=?$

15 16 17 18 14

3 solutions

David Vreken
May 28, 2019

Let $x = \overline{PQRST}$ . Then from the given equation, $(10x + 4)4 = 400000 + x$ , which solves to $x = 10256$ .

Therefore, $P + Q + R + S + T = 1 + 0 + 2 + 5 + 6 = \boxed{14}$ .

CodeCrafter 1
May 28, 2019

$4 \cdot 4 = 16$ Thus $T = 6$

$6 \cdot 4 = 24$ Thus $S = 4 + 1 = 5$ . Don't forget the carry!

$5 \cdot 4 = 20$ Thus $R = 0 + 2 = 2$

$2 \cdot 4 = 8$ Then Q should be 10, but the 1 is the carry for the P's place. Therefore $Q = 0$ .

Then $0 \cdot 4 = 0$ . Thus $P = 0 + 1 = 1$ . Then $4 \cdot 1 = 4$ .

So we get 410256 and the sum is $1 + 0 + 2 + 5 + 6 = 14$ .

Kyle T
May 28, 2019

<?php
for($i=10000;$i<=99999;$i++){
    if(($i.'4')*4 == '4'.$i){
        echo $i.' => '.array_sum(str_split($i));
    }
}
//prints: 10256 => 14
?>