Question

the HCF and LCM of two numbers are 12 and 72 respectively. if the sum of these numbers is 60,then one of the numbers will be

Answer 1

Given:
Let the two numbers be x & y

x+y = 60..............(1)

H.C.F= 12, L.C.M= 72

H.C.F × L.C.M = PRODUCT OF 2 NUMBERS

12 × 72 = XY

864 = XY

X= 864/y..............(2)

Put this value of x in eq 1

x+y= 60

864/y + y= 60

(864+ y²)/y = 60

864 + y² = 60 y

y² -60y +864=0

y² -36y -24y+864=0

y(y-36)-24(y-36)=0

(y-36)(y-24)= 0

y-36= 0

y=36

y-24=0

y=24

Put the value of y= 36

x+y=60

x+ 36= 60

x= 60 -36= 24

x= 24

If y= 36, then x= 24

If y=24, then x= 36

Hence, the numbers are 36 & 24
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Hope this will help you....

Answer 2

Solution :-

Let the two numbers be x ans y respectively.

HCF = 12 and LCM = 72

Then according to the question.

⇒ HCF*LCM = Product of the two numbers

x*y = 12*72

xy = 864 .......(1)

x + y = 60  

x = 60 - y .......(2)

Putting the value of y = 60 - x in (1), we get

(60 - y)*y = 864

60y - y² = 864

⇒ y² - 60y + 864 = 0

⇒ y² - 36y - 24y + 864 = 0

⇒ y(y - 36) - 24(y - 36) = 0

⇒ y - 36 = 0  or y - 24 = 0

⇒ y = 36 or y 24

Putting the value of y in (2)

x = 60 - y            x = 60 - y

x = 60 - 36          x = 60 - 24 

x = 24                 x = 36


If y = 36 then, x = 24

and, if y = 24 then x = 36

So, the two numbers are 36 and 24

Answer