TIFR 2011

Question

A is a two digit number and B is obtained from A by reversing its digits.

For example, if A is 42 then B is 24. Assume that AB is greater than BA (AB is not A multiplied by B. It is just the decimal representation. For example, if A is 42 and B is 24 then AB is 4224 and BA is 2442). Which of the following is always true?

(A) The largest prime factor of AB−BA is 7.

(B) The largest prime factor of AB+BA is 11.

(C) The largest prime factor of AB+BA is 101.

(D) The largest prime factor of AB − BA is always equal to the smallest prime factor of AB+BA

Answer 1

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).