Numbers-Divisibility

Let us discuss divisibility concepts with examples:

Divisibility by 2

Last digit must be even (or Divisible by 2)

Divisibility by 4

Last two digits must be Divisible by 4 (logic here is 4 = 2²)

Divisibility by 8

Last three digits must be Divisible by 8 (logic here is 4 = 2³)

This logic can be extended to any exponent.

Example Problem:

Q1) Check for divisibility of 95076 by 2,4 and 8

As the Last digit 6 is divisible by 2, the given number is divisible by 2

       As the Last two digits 76 is divisible by 4, the given number is divisible by 4

As the Last three digits 076 is not divisible by 8, the given number is not divisible by 8

Divisibility by 3

Sum of digits must be Divisible by 3

Divisibility by 9

Sum of digits must be Divisible by 9

Example Problem:

Q2) Check for divisibility of 95076 by 3 and 9

 Solution:

Sum of the digits = 9+5+0+7+6 =27

As 27 is divisible by 3, the given number is divisible by 3

As 27 is divisible by 9, the given number is divisible by 9

Divisibility by 6

Number must be Divisible by 2 and 3

Divisibility by 12

Number must be Divisible by 4 and 3

Divisibility by 15

Number must be Divisible by 3 and 5

Example Problem:

Q3) Check for divisibility of 86118 by 6 and 12

 Solution:

Checking for divisibility by 6:

As the Last digit 8 is divisible by 2, the given number is divisible by 2

Sum of the digits = 8+6+1+1+8 =24. As 24 is divisible by 3, the given number is    divisible by 3

As the given number is divisible by both 2 and 3, it is divisible by 6 as well.

Checking for divisibility by 12:

As the Last two digits 18 is not divisible by 4, the given number is not divisible by 4 and hence not divisible by 12 as well.

Divisibility by 5,25,125,...

for divisibility by 5, Last digit must be divisible by 5 

for divisibility by 25, Last two digits must be divisible by 25

for divisibility by 125, Last three digits must be divisible by 125

and so on and so forth

Divisibility by 7

Multiply the unit digit with 2 and and subtract that from the remaining part. If the result is divisible by 7 then the given number is also divisible by 7.

Example Problem:

Q4) Check for divisibility of 2583 by 7

 Solution:

1st iteration: unit digit is 3 , 258-2(3)=258 - 6 = 252

2nd iteration:unit digit is 2, 25-2(2) = 25 - 4 = 21

As 21 is divisible by 7, 2583 is also divisible by 7.

Divisibility by 10

Last digit must be 0

Divisibility by 11

Check here whether the difference of sum of digits at odd places and sum of digits at even places is zero or divisible by 11.

 Checking for divisibility by 11:

Q5)1331

Sum of digits at odd places = 1+3 = 4         --- >(1)

Sum of digits at even places = 3+1 = 4        -- > (2)

Difference of (1) and (2) = 4-4 = 0 hence 1331 is divisible by 11

Q6) 9152

Sum of digits at odd places = 2+1 = 3         --- >(1)

Sum of digits at even places = 5+9 =14       -- > (2)

Difference of (1) and (2) = 14-3 = 11, which is divisible by 11. Hence 9152 is divisible by 11.

This concept is quite useful in math-simplifications in general and GRE-word problems and GMAT- DS section problems in particular.