Question

This product of H.C.F and L.C.M Motto number is 24. The difference of the two numbers is 2
Find the numbers.​

Answer 1

Answer:

The product of LCM and HCF of two numbers is always equal to the product of the numbers

Let one number =x

Second number = x+2

Hence x*(x+2)=24

Or x^2+2x-24=0

I.e. (x+6)(x-4)=0

Now either x+6=0 i.e. x=-6

or x-4=0 i.e. x=4

Since the numbers are positive

Hence one numbers=x

And the second number=x+2=4+2= 6

4. and 4+2i.e. 6

4 and 6

Answer 2

Product of HCF and LCM of two numbers is 24.

And the difference of two numbers is 2.

☯ So, Let the two number be x and y.

Therefore,

$\qquad:\implies\sf x - y = 2$

$\qquad:\implies\sf x = 2 + y\qquad\qquad\bigg\lgroup\bf eq.(1)\bigg\rgroup$

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We know that,

$\star\;{\boxed{\sf{\purple{LCM \times HCF = Product\;of\;two\;numbers}}}}$

$\qquad\qquad:\implies\sf LCM \times HCF = x \times y$

$\qquad\qquad\qquad:\implies\sf 24 = x \times y$

$\qquad\qquad\small\sf\dag\; \underline{Putting\;values\;of\;x\;from\;eq\;(1)}$

$\qquad\qquad\quad:\implies\sf 24 = (2 + y) \times y$

$\qquad\qquad\qquad:\implies\sf 24 = (2 + y)y$

$\qquad\qquad\qquad:\implies\sf 24 = 2y + y^2$

$\qquad\qquad\quad:\implies\sf y^2 + 2y - 24 = 0$

$\qquad\quad\quad:\implies\sf y^2 + 6y - 4y - 24 = 0$

$\quad\quad\quad\quad:\implies\sf y(y + 6) - 4(y + 6)= 0$

$\qquad\qquad\quad:\implies\sf (y + 6)(y - 4)= 0$

$\qquad\qquad\quad\quad:\implies\bf y = -6\;,\;y = 4$

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The number can't be negative.

So, y = 4.

☯ Now, Putting values of y in eq(1),

$\qquad:\implies\sf x - 4 = 2$

$\qquad:\implies\sf x = 2 + 4$

$\qquad:\implies\bf y = 6$

$\therefore\;\sf \underline{The\; required\; numbers\;are\;4\;and\;6.}$