VEDIC MATHEMATICS : Divisibility
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Transcript of VEDIC MATHEMATICS : Divisibility
Prasad Divisibility 1
VEDIC MATHEMATICS : Divisibility
T. K. Prasadhttp://www.cs.wright.edu/~tkprasad
Prasad Divisibility 2
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.• 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.
Prasad Divisibility 3
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 4
(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 5
(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 6
(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 7
(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 8
(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 9
(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 10
(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 11
Is 204379 divisible by 7?
204379
=> 20437 – 18
=> 20419
=> 2041 – 18
=> 2023
=> 202 – 6
=> 196
=> 19 – 12
=> 7
Prasad Divisibility 12
(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.