20. Harriet writes a positive integer on each edge of a square. She also writes at each vertex the product of the integers on the two edges that meet at that vertex. The sum of the integers at the vertices is 15 . What is the sum of the integers on the edges of the square?
A 6
B 7
C 8
D 10
E 15
Solution
$\mathbf{C}$

Let the integers written on the edges of the square be $u, v, w$ and $x$ as shown in the diagram. Therefore, since the integer written at each vertex is the product of the integers written on the edges that meet at that vertex, the integers at the vertices are $u x, u v, v w$ and $w x$. Therefore $u x+u v+v w+w x=15$ and hence $(u+w)(v+x)=15$. Since $u, v, w$ and $x$ are positive integers, the smallest either of $u+w$ or $v+x$ can be is $1+1=2$. Therefore the only possible multiplications which give an answer of 15 are $3 \times 5=15$ and $5 \times 3=15$. In each case, the sum of the integers on the edges of the square is $3+5=8$.