Step 1: Convert both lengths into centimetres.
$$4\ \text{m}\ 20\ \text{cm} = 420\ \text{cm},\qquad 5\ \text{m}\ 4\ \text{cm} = 504\ \text{cm}$$
Step 2: To measure both exactly in the least time, the plank must be of the LARGEST possible length, i.e. we need the HCF of $420$ and $504$.
Step 3: Prime factorisation :
$$420 = 2^2 \times 3 \times 5 \times 7$$
$$504 = 2^3 \times 3^2 \times 7$$
Step 4: $$\text{HCF} = 2^2 \times 3 \times 7 = 84$$
Step 5: Hence the length of the plank $= 84$ cm.