Find the length of the plank that can be used to measure the lengths $\displaystyle 4$ m $\displaystyle 20$ cm and $\displaystyle 5$ m $\displaystyle 4$ cm exactly, in the least time.

\(\displaystyle 4 \mathrm{~m} 20 \mathrm{~cm}=420 \mathrm{~cm}\) and \(\displaystyle 5 \mathrm{~m} 4 \mathrm{~cm}=504 \mathrm{~cm}\) Size of plank should be maximum, so we will find HCF \(\displaystyle (420,504)\) \(\displaystyle 420=2^{2} \times 3 \times 5 \times 7\) and \(\displaystyle 504=2^{3} \times 3^{2} \times 7\) HCF \(\displaystyle (420,504)=84\) ∴ the required length of the plank is $\displaystyle 84$ cm