Step 1: Convert both lengths to the same unit.
\[ 4 \text{ m } 20 \text{ cm} = 420 \text{ cm}, \qquad 5 \text{ m } 4 \text{ cm} = 504 \text{ cm} \]
We need the length of the longest plank that measures both exactly, which is the HCF of 420 and 504. Instead of breaking both numbers into prime factors, let's use Euclid's division algorithm, which finds the HCF through repeated division.
Step 2: Apply Euclid's division lemma to the larger and smaller number.
Divide the larger number $504$ by the smaller number $420$:
\[ 504 = 1 \times 420 + 84 \]
The remainder here is $84$, which is not zero, so we continue.
Step 3: Repeat the process, now dividing 420 by the remainder 84.
\[ 420 = 5 \times 84 + 0 \]
This time the remainder is $0$, so we stop here.
Step 4: The last non-zero remainder is the HCF.
By Euclid's division algorithm, the HCF of $420$ and $504$ is the divisor at the step where the remainder became zero, which is $84$.
Final Answer:
The length of the plank that can measure both distances exactly, in the least time, is $84$ cm.
\[ \boxed{84 \text{ cm}} \]