Question

The length, breadth and height of a room are 200 cm, 500 cm and 600 cm respectively.
Find the longest tape which can measure the three dimensions of the room exactly.

Answer 1

$\large\underline{\sf{Solution-}}$

Given Dimensions of room,

• Length = 200 cm

• Breadth = 500 cm

• Height = 600 cm

So,

Let assume that

• The longest tape which can measure the three dimensions of the room exactly be 'x' cm

Thus,

$\bf :\longmapsto\:x = HCF(200, \: 500, \: 600)$

Consider,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 200}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:200\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:100\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:50\: \:\:}}\\{\underline{\sf{5}}}&{\underline{\sf{\:\:25\:\:\:}}} \\ {\underline{\sf{5}}}&{\underline{\sf{\:\:5\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 200 = {5}^{2} \times {2}^{3}}$

Consider,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 500}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:500\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:100\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:50\: \:\:}}\\{\underline{\sf{5}}}&{\underline{\sf{\:\:25\:\:\:}}} \\ {\underline{\sf{5}}}&{\underline{\sf{\:\:5\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 500 = {2}^{2} \times {5}^{3}}$

Consider,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 600}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:600\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:300\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:150\: \:\:}}\\{\underline{\sf{5}}}&{\underline{\sf{\:\:75\:\:\:}}} \\{\underline{\sf{5}}}&{\underline{\sf{\:\:15\:\:\:}}} \\ {\underline{\sf{3}}}&{\underline{\sf{\:\:3\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 600 = {2}^{3} \times {5}^{2} \times 3}$

Thus,

We have now,

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 600 = {2}^{3} \times {5}^{2} \times 3}$

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 500 = {2}^{2} \times {5}^{3}}$

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 200 = {5}^{2} \times {2}^{3}}$

Therefore,

$\rm:\longmapsto\:x = HCF(200, \: 500, \: 600)$

$\bf\implies \:x = {2}^{2} \times {5}^{2} = 100 \: cm$

Hence,

• The longest tape which can measure the three dimensions of the room exactly is 100 cm or 1 m.

Additional Information :-

Let a and b are two natural numbers having hcf as x and lcm as y then

$\boxed{ \bf{ \: x \times y =a \times b}}$

$\boxed{ \bf{ x \leqslant (a,b)}}$

$\boxed{ \bf{ y \geqslant (a,b)}}$

$\boxed{ \bf{ y \: is \: always \: divisible \: by \: x}}$