Question

The difference between two numbers is 15. The lcm and hcf are 180 and 15 respectively. Find the numbers. A. 15,30

Answer 1

Let one number be X then other number will be x+15

as we know that lcm×hcf=product of those numbers

180×15=X(x+15)
2700=x^2+15x
x^2+15x-2700=0
x^2 + 60x - 45x - 2700 =0
X+60 = 0
x-45=0
X=45
One Number = 45
other number = 60
it will not be 15 and 30 as the l.c.m of 15 and 30 will be 30

Answer 2

Given,

The difference between two numbers is 15. The lcm and hcf are 180 and 15 respectively.

To Find,

The numbers =?

Solution,

We can solve the question as follows:

It is given to us that the difference between two numbers is 15. The lcm and hcf are 180 and 15 respectively. We have to find the two numbers.

Let the first number be x and the other number be y.

Assuming that x is greater than y,

$x - y = 15$   ----------- (1)

We know that the product of the LCM and HCF of two numbers is equal to the product of their numbers.

$Product\: of\: two\: numbers = Product\: of\: their\: HCF\: and\: LCM$

Therefore,

$x*y = 180*15\\x*y = 2700$----------- (2)

From the (1) equation,

$x = 15 + y$

Substituting it in the second equation,

$(15 + y)y = 2700$

$15y + y^{2} = 2700$

$y^{2} + 15y - 2700 = 0$

Solving the above quadratic equation,

$y^{2} + 60y - 45y - 2700 = 0$

$y(y + 60) -45(y + 60) = 0$

$(y - 45)(y + 60) = 0$

$y = 45, -60$

Since the number cannot be negative, y is equal to 45.

From equation (1),

$x = 15 + y = 15 + 45 = 60$

Hence, the two numbers are 45 and 60.