Triangle of Nine

We all know how great this number “9” is. It is remarkably known for its cyclic properties which results in beautiful mathematical patterns.

You might also know that,

The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:

The sum of the digits of 41 is 5, and 41 − 5 = 36. The digital root of 36 is 3 + 6 = 9, which demonstrates that it is divisible by nine.

The sum of the digits of 35967930 is 3 + 5 + 9 + 6 + 7 + 9 + 3 + 0 = 42, and 35967930 − 42 = 35967888. The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + 8 + 8 + 8 = 54, 5 + 4 = 9.

Great!. OK lets come back to what I would like you to notice here.

Take any multiple of nine and write it same number of times as the co-factor of 9 is for that number, and add their digital roots, you will get exactly the same number.

Example. Here is a multiple of 9, 18 = 9 * 2 -> 2 is the co-factor of 9 for 18 -> Write 18 for 2 times and Add their digits 18 + 18 -> 9 + 9 -> 18

9 = 9

18 18 =18

27 27 27 =27

36 36 36 36 =36

45 45 45 45 45 =45

54 54 54 54 54 54 =54

63 63 63 63 63 63 63 = 63

72 72 72 72 72 72 72 72 = 72

81 81 81 81 81 81 81 81 81 = 81

90 90 90 90 90 90 90 90 90 90 = 90

If We try this for first 10 positive integral multiples of 9, we will find this beautiful triangle of numbers dominated by 9.

Work by: Piyush Goel

https://piyushtheorem.wordpress.com/2018/03/25/triangle-of-nine/