Question

A room 5m 44cm long and 3m 74cm broad is
to be proved with square tiles. The least
number of square tiles required to cover the
floor is​

Answer 1

Answer:

5984

Step-by-step explanation:

we need to do LCM as it was asked for least tiles required

Answer 2

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

Dimensions of Room :-

• Length of room = 5 m 44 cm = 544 cm

• Breadth of room = 3 m 74 cm = 374 cm

So,

• Area of room is

$\rm :\longmapsto\:Area_{room)} = Length \times Breadth$

$\rm :\longmapsto\:Area_{room)} = 544 \times 374 \: {cm}^{2}$

Now,

• The largest size of square tile = HCF(544, 374)

Now, Consider

• Prime factorization of 544 :-

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:544\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:272\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:136\: \:\:}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:68\:\:\:}}} \\{\underline{\sf{2}}}&{\underline{\sf{\:\:34\:\:\:}}} \\ {\underline{\sf{17}}}&{\underline{\sf{\:\:17\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\bf\implies \:544 = {2}^{5} \times 17$

Prime factorization of 374 :-

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:374\:\:\:}}}\\ {\underline{\sf{11}}}& \underline{\sf{\:\:187\:\:\:}} \\\underline{\sf{17}}&\underline{\sf{\:\:17\: \:\:}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\bf\implies \:374 = 2 \times 11 \times 17$

Hence,

$\rm :\longmapsto\:HCF(544, 374) \: = \: 2 \times 17 = 34$

$\bf\implies \:Size \: of \: square \: tile \: = \: 34 \: cm$

So,

$\rm :\longmapsto\:Area_{square \: tile)} = 34 \times 34 \: {cm}^{2}$

Hence,

$\rm :\longmapsto\:Number _{(tiles)} = \dfrac{Area_{room)}}{Area_{square \: tile)}}$

$\rm :\longmapsto\:Number _{(tiles)} = \dfrac{ \cancel{544} \: \: \: ^{16} \times \cancel{374} \: \: \: ^{11} }{ \cancel{34} \times \cancel{34}}$

$\bf\implies \:Number _{(tiles)} = 176$

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Concept Used :-

The fundamental theorem of arithmetic -

This theorem states,

• "Every composite number can be factorized as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur".

To find the HCF and LCM of two numbers, we prefer the fundamental theorem of arithmetic.

• For this, we first find the prime factorization of both the numbers.

• HCF is the product of the smallest power of each common prime factor.

• LCM is the product of the greatest power of each common prime factor.