Explanation:

Let two positive integers be x and y respectively.

So by statement , 

x-y = 4.........................eq (1)

1/x +1/y = 10/21

(x + y)/xy = 10/21

(x + y)21 = 10xy 

21x +21y = 10xy...........eq (2)

So , from eq (1) 

x = y+4 

Put above equation in eq (2) 

21(y+4) +21y = 10(y+4)y

21y +84 +21y = 40y + 10y^2

-10y^2+2y+84= 0

10y^2 -2y -84 =0

Or 5y^2 -y -42 =0

By solving the quadratic equation we will get the value of y as 

y = 3 , -2.8 

Therefore , x=7 , 1.2 

Bt since it is given in the question that integers should be positive 

Therefore y = 3 and x = 7

Hence, option 1 is the correct answer.