If the sum of two numbers is 1215 and their HCF is 81, then . . .

Question : If the sum of two numbers is 1215 and their HCF is 81, then the possible number of pairs of such numbers are

a) 2

b) 3

c) 4

d) 5

Doubt by Vanshika

Solution : 

HCF of both the numbers is 81, it means that both are divisible by 81 or we can also say that both are the multiples of 81. 
Correct?

Let us consider the two numbers by P and Q
Now 
P = 81x where x is the product of other prime factors.

Q = 81y where y is the product of other prime factors.

Note that x and y both are co-prime [They will not have any common factor other than 1]

ATQ

P+Q = 1215

81x+81y=1215

81(x+y)=1215
x+y=1215/81

x+y=15

So the possible co-prime pairs could be
(1,14),
(2, 13)
(4, 11)

(7, 8)

So the corresponding values of P and Q would be

(81, 1134)

(162, 1053)

(324, 891)

(567, 648)

so total possible pairs of such numbers are 4.

Hence, c) would be the correct option.