Digital Root of Large Numbers
"If we add
the digits of a large number repeatedly till we get a single-digit number, then
this single-digit number is called the digital root of that number."
Let us understand it using a few examples:
Example 1:
Find the
digital root of 4875291.
Step 1: Add the digits of 4875291: 4 + 8 + 7
+ 5 + 2 + 9 + 1 = 36
Step 2: Add the digits of 36: 3 + 6 = 9
Therefore,
the digital root of 4875291 is 9.
Recall the
divisibility test by 9. If the sum of the digits of a number is divisible by 9,
then the number is divisible by 9.
Thus, 4875291
is divisible by 9.
We can say
that the digital root of all multiples of 9 is 9.
Now, add 4
to 4875291. We get 4875295.
4875295 is 4
more than the multiple of 9.
It means if
we divide 4875295 by 9, the remainder is 4.
Now, find
the digital root of 4875295.
4 + 8 + 7 + 5
+ 2 + 9 + 5 = 40
4 + 0 = 4
Thus, the digital
root of 4875295 is 4.
From the above
explanation, we can state the definition of digital root as follows:
"The
digital root of a number is the remainder when the number is divided by 9,
except when the number is a multiple of 9 — in that case, the digital root is 9."
Example
1: Find the digital
root of the following numbers.
a. 3764972 b. 7192640 c. 4201783 d. 5812934
Solution:
a. 3764972
Step 1: 3 + 7 + 6 + 4 + 9 + 7 + 2 = 38
Step 2: 3 + 8 = 11
Step 3: 1 + 1 = 2
Thus, the
digital root of 3764972 is 2.
b.
7192640
Step 1: 7 + 1 + 9 + 2 + 6 + 4 + 0 = 29
Step 2: 2 + 9 = 11
Step 3: 1 + 1 = 2
Thus, the
digital root of 7192640 is 2.
c.
4201783
Step 1: 4 + 2 + 0 + 1 + 7 + 8 + 3 = 25
Step 2: 2 + 5 = 7
Thus, the
digital root of 4201783 is 7.
d.
5812934
Step 1: 5 + 8 + 1 + 2 + 9 + 3 + 4 = 32
Step 2: 3 + 2 = 5
Thus, the
digital root of 5812934 is 5.
Example
2: Find the digital
root of the following.
a. 18x +
6 b. 36y
+ 2
c. 27x + 45y
+ 4 d. 63x + 81y +
1
Solution:
a. 18x +
6
We have, 18x
+ 6 = 9 × 2x + 6
Since 18x is
divisible by 9, the remainder will be 6 if 18x + 6 is divided by 9.
Thus, the
digital root of 18x + 6 is 6.
b. 36y + 12
36y + 12 = 9
× 4y + 9 + 3
= 9(4y + 1) + 3
Here, 9(4y +
1) is divisible by 9.
Thus, if 36y
+ 12 is divided by 9, the remainder is 3.
Therefore,
the digital root of 36y + 12 is 3.
Simply we
can say since 36y is divisible by 9, the digital root of 36y + 12 is the
digital root of 12, that is, 1 + 2 = 3.
c. 27x +
45y + 4
27x + 45y +
4 = 9(3x + 5y) + 4
Here, 9(3x +
5y) is a multiple of 9, so, the digital root of 27x + 45y + 4 is 4.
d. 63x +
81y + 9
63x + 81y + 9
= 9(7x + 9y + 1)
Here, 63x +
81y + 9 is a multiple of 9.
We know that
the digital root of all the multiples of 9 is 9.
Thus, the
digital root of 63x + 81y + 9 is 9.
Related Concepts: