The area of an integer-sided rectangle is $$20$$. What is the minimum possible value of its perimeter?

Let the integer side lengths of the rectangle be $$l$$ and $$b$$, so that the area condition gives

$$l \times b = 20$$

Because $$l,b \in \mathbb{Z^+}$$, list all factor pairs of $$20$$:

$$(1,20),\;(2,10),\;(4,5)$$

For each pair, compute the perimeter $$P = 2(l+b)$$:

• $$l=1,\;b=20 \;\Rightarrow\; P = 2(1+20)=42$$
• $$l=2,\;b=10 \;\Rightarrow\; P = 2(2+10)=24$$
• $$l=4,\;b=5 \;\Rightarrow\; P = 2(4+5)=18$$

The smallest value among $$42, 24,$$ and $$18$$ is $$18$$.

Hence, the minimum possible perimeter is 18.