find a four digit number which get revers when multiply by square of second most positive integer

• If the number is 4-digit, the least significant digit will not be 0, because that would require the most significant digit to be 0 as well. (4*0=0)
  Suitable least-significant/most-significant pairs are
  2/8, the number is 8xy2
  4/6, the number is 6xy4
  6/4, the number is 4xy6
  8/2, the number is 2xy8
  The most-significant digit will cause a carry into the 5th digit when any of the first 3 of these is multiplied by 4. So, the number must be of the form 2xy8.
  
  We can write equations for x and y assuming n is the number carried over into the next place.
  4y+3 = x+10n
  4x+n = y (assuming no carry)
  
  4(4x+n) +3 = x+10n (substitute the second equation into the first)
  16x + 4n +3 = x + 10n (multiply it out)
  15x + 3 = 6n (subtract 4n+x from both sides)
  5x + 1 = 2n (divide both sides by 3)
  
  x must be odd, so possible values are 1,3,5,7,9
  x=1, n=3
  x=3, n=8
  Since n is the carry from the second column to the third, we're pretty sure it won't be 8 or higher. Thus, x=1 and y=(4*1+3)=7
  
  The number you seek is 2178. When multiplied by 4, it is 8712.
  
• 12 years agoHelpfull: Yes(1) No(0)