Denote by R(N) the integer obtained by reversing the digits of a positive integer N.

Determine the largest integer that is certain to divide N⁴ - (R(N))⁴, with N > R(N), regardless of the choice of N.

answer: 99.

x^4-y^4= (x^2-y^2)* (x^2+y^2) and

 (x^2-y^2)= (x-y)* (x+y)*

If

x= 1000a+ 100b+10c+d

and

y=1000d+ 100c+10b+a

x-y  is divisible by 9

x+y  is divisible by 11

ergo  the product is divisible by 99 for any x and y if one is reverse of the other

x^2+y^2 contributes nothing

qed

Posted by Ady TZIDON on 2014-07-22 12:00:14