Question

The sum of two numbers is 135 and their H.C.F. is 27. If their L.C.M. is 162, the numbers are:

• A. 108, 27
• B. 72, 54
• C. 81, 54
• D. 99, 36

Answer 1

The correct option is C

81, 54

H.C.F. of two numbers is 27.
So let the numbers be 27a and 27b
Now, 27a + 27b = 135
$\Rightarrow$ a+ b = 5 .........(i)
Also, product of two numbers = H.C.F. of the numbers $\times$ L.C.M. of two numbers
27a $\times$ 27b = 27 $\times$ 162.
$\Rightarrow$ ab = 6 ............(ii)

Now using the identity:
$(a-b{)}^2=(a+b{)}^2-4ab$
$(a-b{)}^2=5^2-(4\times 6)$
$\Rightarrow$ a - b = 1 ..........(iii)
Solving (i) and (iii) we get
a = 3, b = 2
So, numbers are 27 $\times$ 3 = 81 and 27 $\times$ 2 = 54.