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.