Moving Digits – 3
THE CHALLENGE
Find a 4-digit number, ABCD, that satisfies this interesting equation when you reverse the digits:
ABCD x 9 = DCBA
EXPLORATION
Investigate why this cannot happen for numbers less than 1000. Also, look for numbers larger than 9,999 that have this property.
Notes
THE CHALLENGE
Because multiplying ABCD by 9 produces a 4-digit number, A must be 1.
Our equation is now 1BCD x 9 = DCB1.
D x 9 produces a 1 means that D must be 9.
Our equation is now 1BC9 x 9 = 9CB1.
B must be less than 2 so that 1BC9 x 9 will be less than 9999. C9 x 9 ends in B1 leaves only two possibilities for C – either C is 8 (B = 0) or 7 (B = 1). This gives two possible numbers: 1089 or 1179. Only 1089 works. The answer is 1089.
EXPLORATION
No matter the number of digits, the logic holds that the high-order digit should be 1 and the low-order digit should be 9.
For two-digit numbers, 19 does not work. For three-digit numbers, numbers of the form 1×9 do not work.
The analysis for larger numbers is perhaps more than you’d like to read. Here are the next few numbers beyond 9999: 10989, 109989, and 1099989.