Mathematical Beauties of the Number 76923

Cartesian mathematicism: The long concatenations of simple and easy reasoning which geometricians use in achieving their most difficult demonstrations gave me occasion to imagine that all matters which may enter the human mind were interrelated in the same fashion.
René Descartes

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Numbers are always beautiful and in this blog post let’s play with the beautiful number 76923! One of the occurrences of this number can be found in one–thirteenth (1/13). Let’s start this post with Midy’s theorem and discuss other interesting properties of the number in succession.

Midy’s Theorem If the period of a repeating decimal for a/p, where p is prime and a/p is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. In the case of 1/13 (one–thirteenth), the period is 6. In fact, we have

In this case, we have 076 + 923 = 999.

The prime factorisation of 76923 is 3³ × 7 × 11 × 37.

Divisors of 76923 are 1, 3, 7, 9, 11, 21, 27, 33, 37, 63, 77, 99, 111, 189, 231, 259, 297, 333, 407, 693, 777, 999, 1221, 2079, 2331, 2849, 3663, 6993, 8547, 10989, 25641, and 76923.

76923 is a Harshad number as it is divisible by the sum of its digits:

When we multiply 76923 by other numbers, notice how it yields numbers in the same order but with a different starting point. In fact, the first digit of every product goes to the end of the number to form the next product. Otherwise, the order of the digits is intact. Moreover, the sum of the digits in each of these product numbers is equal to 27.

Sl. No.	Consider the Product	Sum of the digits in the Product Number
1	76923 × 01 = 076923	0 + 7 + 6 + 9 + 2 + 3 = 27
2	76923 × 10 = 769230	7 + 6 + 9 + 2 + 3 + 0 = 27
3	76923 × 09 = 692307	6 + 9 + 2 + 3 + 0 + 7 = 27
4	76923 × 12 = 923076	9 + 2 + 3 + 0 + 7 + 6 = 27
5	76923 × 03 = 230769	2 + 3 + 0 + 7 + 6 + 9 = 27
6	76923 × 04 = 307692	3 + 0 + 7 + 6 + 9 + 2 = 27

Next, notice how various products of 76,923 yield different numbers. Again, the first digit of the product goes to the end of the number to form the next product. Otherwise, the order of the digits is intact. As before, the sum of the digits in each of these product numbers is equal to 27.

Sl. No.	Consider the Product	Sum of the digits in the Product Number
1	76923 × 02 = 153846	1 + 5 + 3 + 8 + 4 + 6 = 27
2	76923 × 07 = 538461	5 + 3 + 8 + 4 + 6 + 1 = 27
3	76923 × 05 = 384615	3 + 8 + 4 + 6 + 1 + 5 = 27
4	76923 × 11 = 846153	8 + 4 + 6 + 1 + 5 + 3 = 27
5	76923 × 06 = 461538	4 + 6 + 1 + 5 + 3 + 8 = 27
6	76923 × 08 = 615384	6 + 1 + 5 + 3 + 8 + 4 = 27

In the above multiplication tables (blue and green), we can see the positive integers from 1 to 12. In fact, it partitions the positive integers from 1 to 12 to into two sets: {1, 3, 4, 9, 10, 12} (first set, blue colour) and {2, 5, 6, 7, 8, 11} (second set, green colour). Is there anything interesting about this division?

We can show that, 6 is the smallest value of n such that 10ⁿ – 1 is divisible by 13. In that case, (10⁶ – 1)/13 = 76923 and we get

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