Question

The H. C. F .and L. C. M.of two numbers are 12 and 72 respectively. If the sum of the two numbers is 60,then one of the two numbers will be

Answer 1

$\large\underline{ \underline{ \sf \: Solution : \: \: \: }}$

Let , the two numbers be x and y

We know that ,

Product of two number = LCM × HCF

= 12 × 72

= 864 ------- eq (i)

Given , $\sf x + y = 60 \: ---- \: eq (ii)$

We know that , (a-b)² = (a+b)² - 4ab

Put the values from eq (i) & eq (ii) , we get ,

$\to \sf {(x - y)}^{2} = {(60)}^{2} - 4 × 864 \\ \\ \to \sf</p><p> {(x - y)}^{2} = 3600 - 3456 \\ \\ \to \sf</p><p> {(x - y)}^{2} = 144 \\ \\ \to \sf</p><p>x - y = 12 \: ---- \: eq (iii)</p><p>$

Add eq (ii) and eq (iii)

$\to \sf 2x = 72 \\ \\ \to \sf</p><p>x = 36$

Put the value of x = 36 in eq (ii)

$\to \sf 36 + y = 60 \\ \\ \to \sf</p><p>y = 24$

Hence , the required numbers are 36 and 24