Question

Two positive integers a and b are such that a + b = a /b + b /a . What is the value of a2 + b2?

Answer 1

a+b = a/b + b/a

Upon calculating the LCM,

a+b=(a2+b2)/ab

 

By cross multiplying

(a+b) ab = a²+b²

 

a².b+b².a = a²+b²

 

By taking a² and b² as common,

a² (b−1) + b² (a−1) = 0

 

Now since a and b are positive integers, the result of their square value cannot be zero.

 

Therefore, to make the above equation equal to zero, (b−1) and (a−1), both have to be 0.

Therefore,

b−1 = 0, i.e., b = 1

 

a−1 = 0, i.e., a = 1

Hence a = b = 1

Answer 2

Answer:

2

Step-by-step explanation:

a+b=a^2+b^2/ab

by cross multiplication

a^2b-a^2+ab^2-b^2=0

a^2(b-1)+b^2(a-1)=0

a and b are +ve integers so, their squares are >0

so,

    b-1=a-1=0

    b=a=1

so,

    a^2+b^2=1^2+1^2=1+1=2