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Let’s look at the first nine multiples of 1,089:

1
089
1 = 1
089
1
089
2 = 2
178
1
089
3 = 3
267
1
089
4 = 4
356
1
089
5 = 5
445
1
089
6 = 6
534
1
089
7 = 7
623
1
089
8 = 8
712
1
089
9 = 9
801

Do you notice a pattern among the products? Look at the first and ninth products. They are the reverses of one another. The second and the eighth are also reverses of one another. And so the pattern continues, until the fifth product is the reverse of itself, known as a palindromic number.?

Notice, in particular, that 1
089
9 = 9
801, which is the reversal of the original number. The same property holds for 10
989
9 = 98
901, and similarly, 109
989
9 = 989
901. Students will be quick to offer extensions to this. Your students should recognize by now that we altered the original 1,089 by inserting a 9 in the middle of the number, and extended that by inserting 99 in the middle of the 1,089. It would be nice to conclude from this that each of the following numbers have the same property: 1,099,989, 10,999,989, 109,999,989, 1,099,999,989, 10,999,999,989, and so on.

? We have more about palindromic numbers in Unit 1.16.

As a matter of fact, there is only one other number with four or fewer
digits where a multiple of itself is equal to its reversal, and that is the number 2,178 (which just happens to be 2
1
089), since 2
178
4 = 8
712. Wouldn’t it be nice if we could extend this as we did with the above example by inserting 9s into the middle of the number to generate other numbers that have the same property? Your students ought to be encouraged to try this independently and try to come to some conclusion. Yes, it is true that

21
978
4 = 87
912
219
978
4 = 879
912
2
199
978
4 = 8
799
912
21
999
978
4 = 87
999
912
219
999
978
4 = 879
999
912
2
199
999
978
4 = 8
799
999
912

Taken From :Math Wonders to inspire teacher and student