Last week we discussed digital sums and the interesting patterns formed by the digital sums of the 5 times table. Shall we try some more patterns?
Look at the 2 times table. The multiples of 2 and the digital sum are as follows:
There are some more tables which also follow the same pattern, but may start from a different number and go along the same path. Look at the 7 times table pattern,
The difference between this and the 2 tables’ pattern is that it starts from 7 and follows the same path. Now you can try 1, 8 and 10 times tables, which will follow the same pattern. Similarly 4 and 5 will have the same pattern. And 3, 6 and 12 will follow the same triangular pattern. There is one very interesting thing and that is the 9 times table. Look at the 9 times table and the digital sums:
Here we see that all the digital sum are 9. So we can come to a conclusion that any multiple of nine always has the digital sum of 9. We also see that adding 9 to any number does not change its digital sum .
Example: 5 + 9 = 14 = 1 + 4 = 5
Therefore we can leave the number 9 out whenever we try to find the digital sum. This is called casting out the nines. And it follows that adding or subtracting any number of 9s will not affect the digital sum of a number. We have discussed this concept earlier, so this is a recap in the context of the digital sum patterns we are discussing now.
To find the digital sum of 1293, we need not add 1 + 2 + 9 + 3 = 15 = 1 + 5 = 6. Just add 1 + 2 + 3 = 6. We get the same answer.
To find the digital sum of 2959 you can cast out the nines and just add 2 and 5. The digital sum is 7. This makes our calculations easier. Not only this, while casting out nines, you can also cast out any group of numbers that add up to 9.
For example, to find the digital sum of 217816, Leave out 2 and 7, 1 and 8. We are left with 1 and 6. 1 + 6 = 7, which is the digital sum of 217816.
Find the digital sum of: 1456, 2793, 85147, 63451