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Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 234 is divisible by
9 because its digit sum = 2 + 3 + 4 = 9 is divisible by 9.
Divisibility by 10

A number is divisible by 10 if its unit digit is 0. For example, 8560 is divisible by 10.
Since, 8560 ends with 0, so 8560 is divisible by 10.
Divisibility by 11

A number is divisible by 11 if the difference between Enrichment
the sum of digits at even places and the sum of digits There is another rule of divisibility by 11.
at odd places is either 0 or a multiple of 11. A number ABCDEF ..... XYZ is divisible by 11

For example: Consider the number 1270324. if Z – Y + X – ... + C – B + A is divisible by 11.
Digit sum of number placed at even places = 2 + 0 For example, 132121 is divisible by 11 because
+ 2 = 4 1 – 2 + 1 – 2 + 3 – 1 = 0 is divisible by 11.
Digit sum of number placed at odd places = 1 + 7 + 3 + 4 = 15

Difference = 15 – 4 = 11
Hence, 1270324 is divisible by 11.

Example 14: Test whether 5281932 is divisible by
(a) 6 (b) 9 (c) 11
Solution: (a) In 5281932, ones digit is 2 so it is divisible by 2.
The sum of digits = 5 + 2 + 8 + 1 + 9 + 3 + 2 = 30, which is divisible by 3. So, the number
5281932 is also divisible by 3.
Since, the given number is divisible by both 2 and 3, so it is divisible by 6 also.

(b) Since, the sum of digits = 30, which is not divisible by 9. So, the number 5281932 is
not divisible by 9.

(c) In 5281932, the sum of digits at odd places (starting from left) = 5 + 8 + 9 + 2 = 24
The sum of digits at even places (starting from left) = 2 + 1 + 3 = 6
The difference between two sums = 24 – 6 = 18, which is not divisible by 11.

Hence, the number 5281932 is not divisible by 11.
Example 15: Write the smallest and the greatest possible digit in the blank space of the number
4765_2 so that the number formed is divisible by 3.
Solution: In 4765_2, sum of digits = 4 + 7 + 6 + 5 + _ + 2 = 24 + _
Multiples of 3 greater than or equal to 24 are 24, 27, 30, 33, 36, ...
Number would be divisible by 3 when 24 + _ = 24, 27, 30 or 33
So, the possible digits to be put in the blank are 0, 3, 6 and 9.

Hence, the smallest digit = 0 and the greatest digit = 9.
Example 16: Replace * with the suitable digit so that the number 7254 * 98 would be divisible by 11.
Solution: In 7254 * 98, sum of digits at odd places (starting from left) = 7 + 5 + * + 8 = 20 + *

The sum of digits at even places (starting from left) = 2 + 4 + 9 = 15

155 Prime Time