What is the number of integral solutions of the equations

What is the number of integral solutions of the equations HCF(a,b) = 5 and a + b = 65?

  1. Exactly one
  2. Infinitely many
  3. Less than 65
  4. None

Answer

Since HCF (a,b) = 5, let a = 5x and b = 5y

So, 5x + 5y = 65

x + y = 13

So, possible pairs (x,y) are (1,12), (2,11), (3,10), (4,9), (5,8), and (6,7).

So, there are 6 solutions which is less than 65.

The correct option is C.