This document provides information about calculating digital roots of numbers. It discusses how to calculate the digital root by summing the digits of a number repeatedly until a single digit is obtained. Examples are provided to illustrate calculating the digital root of various numbers. Properties of digital roots are also outlined, such as the digital root of a square being 1, 4, 7, or 9 and the digital root of a prime number (except 3) being 1, 2, 4, 5, 7, or 8. Justification is given for why prime numbers have these possible digital roots. The document concludes by discussing how to find the last digit of the square root of a number based on the last digit of the number.

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HARSHIT GUPTA
B.C.A. –FIRST YEAR
SECOND SEMESTER
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2. 2
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3. Lets learn
multiplication of 2 digit
numbers.
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4. Suppose we have to
multiply 97 X 98 :
We should take 100
as the base of our
calculation because
it is the nearest
unity to the
numbers to be
multiplied.
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5. For
Example
97-02
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6. Now lets learn
multiplication above the
base.
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7. Suppose we have to
multiply 12 X 14 :
We should take 10
as the base of our
calculation because
it is the nearest
unity to the
numbers to be
multiplied.
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8. For
Example
CLICK HERE TO GO TO THE MAP
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9. 9
D
Here we come to
know about
Digital root

10. Definition
• Digital root of a number is the single
digit obtained by repeatedly summing all
the digits of a number.
• Example:
• Digital root of 2357 = 8
because (2 + 3 + 5 + 7 = 17) and (1 + 7 = 8)
• Digital root of 89149 = 4
because (8 + 9 + 1 + 4 + 9 = 31) and (3 + 1 = 4)
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11. Some facts we already
know
• A number is divisible by 3 if its
digital root is divisible by 3 (that is, it
is 0, 3, 6, or 9).
• 1236 is divisible by 3 because 3 is divisible
by 3.
• Note (1+2+3+6 = 12) and (1+2 = 3).
• Recall: 1x(999+1) + 2x(99+1) + 3x(9+1) + 6
• A number is divisible by 9 if its
digital root is divisible by 9 (that is, it
is 0 or 9). 11

12. (cont’d)
• The digital root of a number is the
remainder obtained by dividing it by 9.
• 1236 divided by 9 = … R 3
• Recall: 1x(999+1) + 2x(99+1) + 3x(9+1) + 6
• Note that 9 is treated similar to 0.
• 36 divided by 9 = … R 0
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13. Properties of digital roots
• Digital root of a square is 1, 4, 7, or 9
• Digital root of a perfect cube is 1, 8 or 9
• Digital root of a prime number (except
3) is 1, 2, 4, 5, 7, or 8
• Digital root of a power of 2 is 1, 2, 4, 5,
7, or 8
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14. Justification for digital
roots of a prime number
• Recall that digital root of 0, 3, 6, or 9
implies it is divisible by 3.
• The digital root of 1, 2, 4, 5, 7, and 8 are
realizable by the prime numbers 19, 2
(11), 13, 5 (23), 7 (43), and 8 (17),
respectively.
– This is a necessary (but not sufficient)
condition for a number to be prime.
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Product of both digital
root then find
Digital root
Digital root of
answer
Digital root of
first number

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7*8=>56=5+6
=>11=>1+1=>2
25
2+5
=7
26
=> 2+6
=>8
25*26=650=>6+5+0
=>11=>1+1=>2

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Last Digit of Perfect
Square Root number
Last digit of root
1 1 Or 9
4 2 Or 8
5 5
6 4 Or 6
9 3 Or 7
Find the Square root of 5041
now 5041 1
Then √50>=7
Then we have to option 71 or 79
Check than we say √5041=71 9
Square
Root ?

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