9 -A Magic Number

9 is the largest 1-digit number and it has many unique properties.

We have already studied Nine Point Circle and it’s benefits in finding the digit sum.

  • The digit sum does not change by adding  9 or subtracting 9 from it. 
  • The digit sum of any multiple of 9 is always 9.

Look at the table of 9:

9 × 1   =      9
9 × 2   =   1 8
9 × 3   =   2 7
9 × 4   =   3 6
9 × 5   =   4 5
9 × 6   =   5 4
9 × 7   =   6 3
9 × 8   =   7 2
9 × 9   =   8 1
9 × 10 =   9 0
9 × 11 =   9 9
9 × 12 = 10 8

On the right side of all products in every case, the digit sum is 9.
There is a unique observation in the above products. If you read the answers as two columns the left column goes up (1, 2, 3, . . .) and the right column goes down
(9, 8, 7, . . .).
This makes it easy to get the answers in the table of 9.

There is no need to remember the table of 9. We can use our fingers to multiply by nine.
Suppose the fingers of your hands are numbered as shown below:

table-9
For example, multiply 3 by 9, simply fold down the 3rd finger.
You will find 2 fingers to the left of the folded finger and 7 fingers to the right.
So 3 × 9 = 27.

Why don’t you practice with your fingers and have fun!

Digit Sum

What is a digit?

“Digit” is derived from Latin word digitus meaning ‘finger, toe’. Digit means any numeral below 10, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

And “Sum” means to add.

A “Digit Sum” is the sum of all of the digits of a number, and is found by adding all of the digits of a number.

For example:

  1. The Digit Sum of 32 is 3+2 = 5
  2. The Digit Sum of 152 is 1+5+2 = 8

If the sum of the digits is greater than 9, then sum the digits of the result again until the result is less than 10.

For example:

  1. The Digit Sum of 57 is 5+7 = 12 → 1+2 = 3 ….. Here 12 >9, so sum the digits again.
    So the Digit Sum of 57 is 3.
  2. The Digit Sum of 687 is 6+8+7 = 21 → 2+1 = 3…. Here 21 >9, so sum the digits again.
    So the Digit Sum of 687 is 3.

Keep finding the Digit Sum of the result until it’s less than 10. Also, in case of digit sum, 0 and 9 are equivalent!

Why is digit sum important?

Digit sum is used for checking calculations, and this becomes highly useful where large numbers are involved. I will explain this part in subsequent sections.

In this course, we are going to study about below topics:

  • Nine Point Circle
  • Casting Out nines (9s)
  • Checking with Digit Sums
  • A magic number

Divisibility Check

Division is opposite of multiplication, and is repeated subtraction.

There is a unique pattern in division, that will help you to find whether any number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 or so on.

In this topic, we are going to learn how to check if a number is divisible by any of the above numbers without actually dividing it.

Let us compare the Traditional way of divisibility check with Vedic way.

Is 24567890 is divisible 3?

Traditional way:

The traditional way is very lengthy and takes time to check divisibility.

division traditional.PNG

Vedic Way:

Check if sum of all digits of this number is divisible by 3.

Sum of digits = 2+4+5+6+7+8+9+0 = 41

Since 41 is not divisible by 3, so number 24567890  is also not divisible by 3.

In this topic, we are going to cover divisibility check by numbers 2, 3, 4, 5, 6, 7, 8, 9, and 10.