Question

If the sum of squares of two numbers is 2754 and their hcf and lcm are 9 and 135 then the numbers are

Answer 1

x² + y² = 2754

HCF = 9 = 3 x 3

LCM = 135 = 3 x 3 x 3 x 5

x = 3 x 3 x 3 = 27

y = 3 x 3 x 5 = 45

27² + 45² = 2754

Answer 2

Let the number = x and y

Product of two numbers = HCF x LCM

$x \times y = 9 \times 135$

$xy = 1215 \: \: \: \: \: \: equation \: (1)$

Given ,

That sum of the squares of two numbers is 2754

${x}^{2} + {y}^{2} = 2754$

$(x + y) ^{2} = {x}^{2} + {y}^{2} + 2xy$

$(x + y) ^{2} = 2754 + 2(1215) = 5184$

$x + y = \sqrt{5184} = 72$

$x + y = 72 \: \: \: \: \: \: equation(2)$

Solving the value of "y" -

$x = 72 - y$

$substitute \: value \: of \: x \: \: in \: equation \: (1)$

$(72 - y)y = 1215$

$72y - {y}^{2} = 1215$

${y}^{2} - 72y + 1215 = 0$

$(y - 27)(y - 45) = 0$

$y = 27 \: or \: 45$

$substitute \: value \: of \: \: y \: \: in \:equation \: (1)$

$xy = 1215$

$x(45) = 1215$

$x = \dfrac{1215}{45} = 27$

$x = 27$

Hence :. x = 27 , y = 45 .

$\bold{Answer: \: 27 \: and \: 45}$