Question

If the HCF of the two numbers be 13 and lcm 273 then the sum of the numbers will be?

Answer 1

Let the two numbers be a and b.

Since the H.C.F. is 13, we can write

a=13xa=13x

b=13yb=13y

for any two positive integers x and y.

Given L.C.M. = 273. We can write

13×x×y=27313×x×y=273

⟹xy=273/13⟹xy=273/13

xy=21xy=21

Given a>13a>13and b>13b>13,

13x>13⟹x>113x>13⟹x>1

13y>13⟹y>113y>13⟹y>1

Now, xy=21xy=21

We need to find x and y.

So, factors of 21 are 1,3,7 and 21.

(x,y) can't be (1,21) or (21,1) since x and y should be greater than 1.

Therefore, (x,y) = (3,7) or (7,3)

Thus,

a = 13×3 = 39

b = 13×7 = 91

Thus, (a,b) = (39,91) or (91,39)

Sum of a and b = (a+b) = 39+91 = 130