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
