A rectangular courtyard measuring 105\text{ cm} in length and 135\text{ cm} in width is to… — State PSC Exam

A rectangular courtyard measuring $105\text{ cm}$ in length and $135\text{ cm}$ in width is to be completely paved with identical square tiles without cutting any tile. What is the maximum possible side length of each square tile?

$15\text{ cm}$ Cevap
B
$5\text{ cm}$
C
$45\text{ cm}$
D
$945\text{ cm}$

Cevap

The maximum possible side length of each square tile is

15\text{ cm}

The maximum side length of a square tile that fits exact integer counts along both length and width is given by

\text{HCF}(105, 135)

. By prime factorization,

105 = 3 \times 5 \times 7

and

135 = 3^3 \times 5

. The common prime factors are

3

and

5

, each with lowest power

1

. Therefore,

\text{HCF} = 3 \times 5 = 15\text{ cm}

Adım Adım Çözüm

Identify the required mathematical operation

To find the maximum side length of identical square tiles that fit whole dimensions of

105\text{ cm}

and

135\text{ cm}

, find the Highest Common Factor (HCF) of

105

and

135

The side of each square tile must divide both the length and the width completely, and to maximize tile size, we need the largest such divisor.

Perform prime factorization of both numbers

105 = 3 \times 5 \times 7

135 = 3^3 \times 5 = 3 \times 3 \times 3 \times 5

Expressing numbers as products of prime factors allows easy identification of common factors.

Calculate the HCF using prime powers

\text{HCF}(105, 135) = 3^1 \times 5^1 = 15

HCF is obtained by taking the product of the smallest powers of all common prime factors.

Anahtar Kavram

Application of HCF and Prime Factorization in Spatial Tiling Problems

Tahmini Süre:45s

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