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VEDIC MATHEMATICS : Divisibility. T. K. Prasad http://www.cs.wright.edu/~tkprasad. 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

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VEDIC MATHEMATICS : Divisibility

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Vedic mathematics divisibility l.jpg

VEDIC MATHEMATICS : Divisibility

T. K. Prasad

http://www.cs.wright.edu/~tkprasad

Divisibility


Divisibility l.jpg

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.

Divisibility


Divisibility by numbers l.jpg

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.

Divisibility


Cont d l.jpg

(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!

Divisibility


Cont d5 l.jpg

(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!

Divisibility


Cont d6 l.jpg

(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!

Divisibility


Cont d7 l.jpg

(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 …!

Divisibility


Cont d8 l.jpg

(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, …!

Divisibility


Cont d9 l.jpg

(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, …!

Divisibility


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(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

Divisibility


Is 204379 divisible by 7 l.jpg

Is 204379 divisible by 7?

204379

=> 20437 – 18

=> 20419

=> 2041 – 18

=> 2023

=> 202 – 6

=> 196

=> 19 – 12

=> 7

Divisibility


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(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 if20x+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.

Divisibility


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