24. A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45 ?
A 9180
B 9090
C 9000
D 8910
E 8190
24. B For a number to be a multiple of 45 it must be a multiple of 5 and also of 9 . In order to be a multiple of 5 , a number's units digit must be 0 or 5 . However, the units digit of a palindromic number cannot be 0 , so it may be deduced that any palindromic number which is a multiple of 45 both starts and ends in the digit 5 . In order to make the desired number as large as possible, its second digit should be 9 and for it to be as small as possible its second digit should be 0 . So, if possible, the numbers required are of the form ' $59 x 95$ ' and ' $50 y 05$ '. In addition, both numbers are to be multiples of 9 which means the sum of the digits of both must be a multiple of 9 . For this to be the case, $x=8$ and $y=8$, giving digit sums of 36 and 18 respectively. So the two required palindromic numbers are 59895 and 50805. Their difference is 9090 .