Back To Front
Number Theory
Level
3
Find a four-digit number that when multiplied by 9, gives the number in reverse.
The answer is 1089.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
A must be 1 as to not have a 5 digit answer, and so must D be 9. Since there were no carrying over to the thousands place, B must be 0. Then 9C + 8 (from the tens of 9 x 9) equal some tens (ones digit equal B). Thus C must be 8. The answer ABCD = 1089.
Or for the number theory approach :
9(1000a + 100b + 10c + d) = 1000d + 100c + 10b + a
8999a + 890b = 10c + 991d
Notice that both b & c cannot contribute to ones place, and because a cannot be 0 (or the number would not be a 4 digit number), d can only be 9 if only to 'catch up' to the least of LHS. That makes the RHS to end in a 9 (0 from 10c plus 9 from 991 x 9), so a being the one directly affected, has to be 1 to equal RHS's ones digit of 9.
8999 + 890b = 10c + 8919
80 + 890b = 10c
8 + 89b = c
Because c is a one digit number as the way we set up the equation earlier, b must be 0, and c equal 8.
So answer abcd = 1089.