I was requested for homework to find how many four-digit numbers exist with the property of an exact division with their reverse ones, not palindromic(eg.1991,1001,2332) and with the first and last digit not 0. I have to send it till thursday. I appreciate any help and I warn you that it is a difficult problem, so don't try to solve it if you have little mathematics experience. Thanks in advance.
Spoiler: I found out that there are 2 such numbers by running an algorithm in Pascal. But I want the mathematical proof.......
Spoiler: I found out that there are 2 such numbers by running an algorithm in Pascal. But I want the mathematical proof.......
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9801 and 8712 have this property.
Simply doing the division (9801/1089 = 9 and 8712/2178 = 4) IS the "mathematical proof".
Your Pascal algorithm IS the "mathematical proof" that these are the only two cases.
[Incidentally, here's something amazing related to your question:
1089x9 = 9801
10989x9 = 98901
109989x9 = 989901
1099989x9 = 9899901 etc.
2178x4 = 8712
21978x4 = 87912
219978x4 = 879912
2199978x4 = 8799912 etc.]
Simply doing the division (9801/1089 = 9 and 8712/2178 = 4) IS the "mathematical proof".
Your Pascal algorithm IS the "mathematical proof" that these are the only two cases.
[Incidentally, here's something amazing related to your question:
1089x9 = 9801
10989x9 = 98901
109989x9 = 989901
1099989x9 = 9899901 etc.
2178x4 = 8712
21978x4 = 87912
219978x4 = 879912
2199978x4 = 8799912 etc.]