let x and y be two digits.
given – A = 10x + y and B = 10y + x
therefore, AB = (10x+y)*100 + (10y+x) = (1000+1)x + (100+10)y
similarly, BA = (1000+1)y + (100+10)x
therefore AB + BA = 1000(x+y) + 100(x+y) + 10(x+y) + (x+y) = (1111)(x+y) = (101)(11)(x+y)
therefore AB – BA = 1000(x-y) + 100(y-x) + 10(y-x) + (x-y) = (891)(x-y) = (3)^4 * (11) (x-y)
option A can’t be true since largest prime factor of AB – BA is 11.
option B can’t be true since largest prime factor of AB + BA is 101.
option C is true.
option D can’t be always true since smallest prime factor of AB + BA can sometimes be (x+y).