VEDIC MATHEMATICS : Divisibility

1
VEDIC MATHEMATICS Divisibility

• T. K. Prasad
• http//www.cs.wright.edu/tkprasad

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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.

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.

4
(contd)

• 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!

5
(contd)

• 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!

6
(contd)

• 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!

7
(contd)

• 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 (991) q (91) r
• 9 and 99 are divisible by 3.
• Is 2832 divisible by 3?
• YES because (283215) is, (156) is !

8
(contd)

• 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 (991) q (91) r
• 9 and 99 are divisible by 9.
• Is 12348 divisible by 9?
• YES, because (1234818) is, (189) is, !

9
(contd)

• 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 (991) q (11-1) r
• 11 and 99 are divisible by 11.
• Is 12408 divisible by 11?
• YES, because (1-24-0811) is, (1-10) is, !

10
(contd)

• 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 gt 35 27 gt 21

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Is 204379 divisible by 7?

• 204379
• gt 20437 18
• gt 20419
• gt 2041 18
• gt 2023
• gt 202 6
• gt 196
• gt 19 12
• gt 7

12
(contd)

• Informal Justification
• A multi-digit number is 10xy (e.g., 176 is
  17(10)6).
• 10xy is divisible by 7 if and only if 20x2y is
  divisible by 7. (2 and 7 are relatively prime).
• Subtracting 20x2y from 21x does not affect its
  divisibility by 7, because 21 is divisible by 7.
• But (21x 20x 2y) (x 2y).
• So (10xy) is divisible by 7 if and only if
  
  
  (x-2y) is divisible by 7.