Question

consider the two numbers whose sum is 135 and their hcf is 27 if their lcm is 162 then what will be the larger number​

Answer 1

SOLUTION

GIVEN

• Consider the two numbers whose sum is 135

• Their HCF is 27.

• Their LCM is 162

TO DETERMINE

The larger number

EVALUATION

Here it is given that the HCF of two given numbers are 27

Let the numbers are 27a and 27b ( a > b )

The sum of the numbers = 135

⇒ 27a + 27b = 135

⇒ a + b = 5 - - - - - (1)

Now it is given that LCM = 162

HCF × LCM = Product of the numbers

⇒ 27 × 162 = 27a × 27b

⇒ 27a × 27b = 27 × 162

⇒ ab = 6 - - - - - - (2)

From Equation 1 and Equation 2 we get

a( 5 - a) = 6

$\displaystyle \sf{ \implies 5a - {a}^{2} = 6}$

$\displaystyle \sf{ \implies {a}^{2} - 5a + 6 = 0}$

$\displaystyle \sf{ \implies {a}^{2} - 3a - 2a + 6 = 0}$

$\displaystyle \sf{ \implies a(a - 3) - 2(a - 3) = 0}$

$\displaystyle \sf{ \implies (a - 3) (a - 2) = 0}$

a - 3 = 0 gives a = 3

a - 2 = 0 gives a = 2

From Equation 1

When a = 3 we have b = 2

When a = 2 we have b = 3

We have assumed a > b

∴ a = 3 , b = 2

Consequently the required numbers are 81 and 54

FINAL ANSWER

Hence the required larger number = 81

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