Question

two positive integers a and b such that a+b=ab+ba,then find the value of a^2+b^2?

Answer 1

a+ b= ab
squaring both sides we get.

a^ 2 + b^2+2 ab=(ab)^2
=> a^ 2 + b^2= (ab)^2-2ab.

Answer 2

a+b=a/b+b/aa+b=a/b+b/a

-> Now take L.C.M

a+b=(a2+b2)/aba+b=(a2+b2)/ab

-> Cross multiplying

(a+b)ab=a2+b2(a+b)ab=a2+b2

a2.b+b2.a=a2+b2a2.b+b2.a=a2+b2

Take a2a2 and b2b2 common

a2(b−1)+b2(a−1)=0a2(b−1)+b2(a−1)=0 --(1)

Now since aa and bb are positive integers - their square can’t be zero.

So in order to make the equation 1 equal to zero:

(b−1)(b−1) and (a−1)(a−1)both has to be 00.

Therefore,

b−1=0b−1=0 =>b=1b=1

and,

a−1=0a−1=0 =>a=1a=1

Hence a=b=1