142,857 . 2 = 285,714
142,857 . 3 = 428,571
142,857 . 4 = 571,428
142,857 . 5 = 714,285
142,857 . 6 = 857,142
You can see symmetries in the products. Notice also that the same digits are used in the product as in the first factor. Furthermore the order of the digits are in the same sequence.
Now look at the product, 142,857 . 7 = 999,999.
And 142,857 . 8 = 1,142,856. If we remove the millions digit and add it to the units digit, the original number is formed.
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills – the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists – the authors themselves rely heavily on it! – but for serious users of mathematics in virtually every discipline.
Various products of 76,923 yield numbers in the same order but with a different starting point.
76,923 . 1 = 076,923
76,923 . 10 = 769,230
76,923 . 9 = 692,307
76,923 . 12 = 923,076
76,923 . 3 = 230,769
76,923 . 4 = 307,692
Below, various products of 76,923 yield different numbers from those above.
76,923 . 2 = 153,846
76,923 . 7 = 538,461
76,923 . 5 = 384,615
76,923 . 11 = 846,153
76,923 . 6 = 461,538
76,923 . 8 = 615,384
Interesting patterns can be obtained by multiplying those numbers.
1 . 1 = 1
11 . 11 = 121
111 . 111 = 12321
1111 . 1111 = 1234321
11111 . 11111 = 123454321
111111 . 111111 = 12345654321
1111111 . 1111111 = 1234567654321
11111111 . 11111111 = 123456787654321
111111111 . 111111111 = 12345678987654321
1 . 8 + 1 = 9
12 . 8 + 2 = 98
123 . 8 + 3 = 987
1234 . 8 + 4 = 9876
12345 . 8 + 5 = 98765
123456 . 8 + 6 = 987654
1234567 . 8 + 7 = 9876543
12345678 . 8 + 8 = 98765432
123456789 . 8 + 9 = 987654321