Pattern in numbers

Question

I bumped into this mathematical calculation while driving my car. With lot of traffic jams and lot of time to kill, I did some scrambling of the registration numbers of the cars in front of me.

I came across a rather strange pattern that I could not relate to any law in mathematics.

For a given number, partition it into any number of partition and where each partition can be any combination of digits from that number. For example

1. Take any number - say, 5437,

2. partition it in any two sets, say 57 and 43 (Other valid partition sets will be 543/7, 37/45, 347/5, 3/4/37, 3/4/7/5)

3. add those partition - 57 + 43 = 100

4. repeat 2 and 3 till 3 comes to single digit. In this case 100 is partitioned into 10/0 : 10 + 0 = 10, 10 => 1/0 : 1 + 0 = 1

So 5437 => 1 (Result 1)

For same number, choose any other partition

2. 543 and 7

3. 543+ 7 = 550

4. 550 partitioned in to 5/50

5. 5 + 50 = 55, partition 55 into 5/5 => 5+5 = 10, partition 10 into 1 + 0 = 1 (same as result 1)

For same number, choose any other partition

2. 537/ 4

3 537 + 4 = 541

4, 541 : 45 /1

5. 45 + 1 = 46, 46=> 4/6 : 4 + 6 = 10, 10=> 1/0 1 + 0 = 1 (Result 1)

Apply any combination of partition, for this number it will be always 1.

Lets take any other number - 91522

1. Partition - 922 and 15, 922 + 15 = 937, Partition 937 -> 73 and 9, 73 + 9 = 82, 82 -> 8 + 2 = 10, 10 : 1 + 0 = 1

2. Partition - 522 and 91, 522 + 91 = 613, Partition 613 -> 63 and 1, 63 + 1 = 64, 64 -> 6 + 4 = 10, 10 -> 1 + 0 = 1

One more 4443

1. Partition - 344 and 4, 344 + 4 = 348, Partition 348 -> 84 and 3, 84 +3 = 87, 87 -> 8 + 7 = 15, 15 : 1 + 5 = 6

2. Partition - 44 and 43, 44 + 43 = 87, Partition 87 -> 8 + 7 = 15, 15 : 1 + 5 = 6

3 Partition - 4 and 443, 4 + 443 = 447, Partition 447 -> 74 + 4 = 78, 78 : 7 + 8 = 15, 15: 5 + 1 = 6

4. Partition - 4, 4, 4, 3, 4 + 4+ 4+3 = 15, 15: 5 + 1 = 6

How can this be explained?

Answer 1

As Doug M said what you are doing is obtaining the modulo $9$ of the number, it is also called the Digital root of the number.

An excerpt from Wikipedia:

The digital root (also repeated digital sum) of a non-negative integer is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.

It helps to see the digital root of a positive integer as the position it holds with respect to the largest multiple of 9 less than it. For example, the digital root of 11 is 2, which means that 11 is the second number after 9. Likewise, the digital root of 2035 is 1, which means that 2035 − 1 is a multiple of 9. If a number produces a digital root of exactly 9, then the number is a multiple of 9.

You can also review the divisibility rules of $9$ here for more details about the rules behind this behavior.