VEDIC MATHEMATICS :
Divisibility
T. K. Prasad
http://www.cs.wright.edu/~tkprasad
Prasad
Divisibility
1
Divisibility
• A number n is divisible by f if there exists
another number q such that n = f * q.
– f is called the factor and q is called the
quotient.
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Prasad
25 is divisible by 5
6 is divisible by 1, 2, and 3.
28 is divisible by 1, 2, 4, 7, 14, and 28.
729 is divisible by 3, 9, and 243.
Divisibility
2
Divisibility by numbers
• Divisibility by 1
– Every number is divisible by 1 and itself.
• Divisibility by 2
– A number is divisible by 2 if the last digit is
divisible by 2.
• Informal Justification (for 3 digit number):
pqr = p * 100 + q * 10 + r
Both 100 and 10 are divisible by 2.
Prasad
Divisibility
3
(cont’d)
• Divisibility by 4
– A number is divisible by 4 if the number
formed by last two digits is divisible by 4.
• Informal Justification (for 3 digit number):
pqr = p * 100 + q * 10 + r
100 is divisible by 4.
• Is 2016 a leap year?
• YES!
Prasad
Divisibility
4
(cont’d)
• Divisibility by 5
– A number is divisible by 5 if the last digit is 0
or 5.
• Informal Justification (for 4 digit number):
apqr = a * 1000 + p * 100 + q * 10 + r
0, 5, 10, 100, and 1000 are divisible by 5.
• Is 2832 divisible by 5?
• NO!
Prasad
Divisibility
5
(cont’d)
• Divisibility by 8
– A number is divisible by 8 if the number
formed by last three digits is divisible by 8.
• Informal Justification (for 4 digit number):
apqr = a * 1000 + p * 100 + q * 10 + r
1000 is divisible by 8.
• Is 2832 divisible by 8?
• YES!
Prasad
Divisibility
6
(cont’d)
• Divisibility by 3
– A number is divisible by 3 if the sum of all the
digits is divisible by 3.
• Informal Justification (for 3 digit number):
pqr = p * (99+1) + q * (9+1) + r
9 and 99 are divisible by 3.
• Is 2832 divisible by 3?
• YES because (2+8+3+2=15) is, (1+5=6) is …!
Prasad
Divisibility
7
(cont’d)
• Divisibility by 9
– A number is divisible by 9 if the sum of all the
digits is divisible by 9.
• Informal Justification (for 3 digit number):
pqr = p * (99+1) + q * (9+1) + r
9 and 99 are divisible by 9.
• Is 12348 divisible by 9?
• YES, because (1+2+3+4+8=18) is, (1+8=9) is, …!
Prasad
Divisibility
8
(cont’d)
• Divisibility by 11
– A number is divisible by 11 if the sum of the
even positioned digits minus the sum of the odd
positioned digits is divisible by 11.
• Informal Justification (for 3 digit number):
pqr = p * (99+1) + q * (11-1) + r
11 and 99 are divisible by 11.
• Is 12408 divisible by 11?
• YES, because (1-2+4-0+8=11) is, (1-1=0) is, …!
Prasad
Divisibility
9
(cont’d)
• Divisibility by 7
– Unfortunately, the rule of thumb for 7 is not
straightforward and you may prefer long
division.
– However here is one approach:
• Divisibility of n by 7 is unaltered by taking the last
digit of n, subtracting its double from the number
formed by removing the last digit from n.
• 357 => 35 – 2*7 => 21
Prasad
Divisibility
10
Is 204379 divisible by 7?
204379
=> 20437 – 18
=> 20419
=> 2041 – 18
=> 2023
=> 202 – 6
=> 196
=> 19 – 12
=> 7
Prasad
Divisibility
11
(cont’d)
• Informal Justification
– A multi-digit number is 10x+y (e.g., 176 is
17*(10)+6).
– 10x+y is divisible by 7 if and only if 20x+2y is
divisible by 7. (2 and 7 are relatively prime).
– Subtracting 20x+2y from 21x does not affect its
divisibility by 7, because 21 is divisible by 7.
– But (21x – 20x – 2y) = (x – 2y).
– So (10x+y) is divisible by 7 if and only if
(x-2y) is divisible by 7.
Prasad
Divisibility
12