Number System - 2

After types of number system, we study about divisibility property of numbers.

There are certain tests for divisibility of numbers by any of the numbers 2, 3, 4, 5, 6, 8, 9, 10 and 11 such that by simply examining the digits in the given number, one can easily determine whether or not a given number is divisible by any of these numbers.

Some test detailed as follows:

Divisibility by 2
If the last digit is an even number or it has zero (0) at the end.
Ex.:- 74, 148, 1210 are all divisible by 2.

Divisibility by 3
If the sum of the digits of the given number is divisible by 3, then that number is fully divisible by 3.
Ex.:- The sum of the digits of number 3705 is 3 + 7 + 0 + 5=15. 15 is divisible by 3 so the number 3705 is fully divisible by 3.

Divisibility by 4
If the formed by the last two digits of the given number is divisible by 4, or if the last two digits are ‘00’, then that number is fully divisible by 4.
Ex.:-216560 is a number whose last two digits are 60, 60 is divisible by 4 so the given number 216560 is also divisible by 4.

Divisibility by 5
If the last digit of the given number is 0 or 5 then the number is fully divisible by 5.
Ex.:- 865, 1605, 5670 are all divisible by 5.

Divisibility by 6
If the given number is divisible by 2 and 3 then that number is also divisible by 6.
Ex.:- let us consider the number is 89004. It has 4 as the last digit, so it is divisible by 2.
Sum of the digits 8 + 9 + 0 + 0 + 4=21, 21 is divisible by 3.
Since, 89004 is divisible by 2 and 3 both so that number is also divisible by 6.

Divisibility by 8
If the number formed by the last three digits of the given number is divisible by 8 or if the last three digits are ‘000’, then that number is fully divisible by 8.
Ex.:- The number 56976 has 976 as the last three digits. Since 976 is divisible by 8, 56976 is also divisible by 8. The number 5463000 end with ‘000’ and so, it is divisible by 8.

Divisibility by 9
If the sum of the digits of the given number is divisible by 9, then that number is also fully divisible by 9.
Ex.:- 890676 is a number the sum of whose digits is = 8 + 9 + 0 + 6 + 7 + 6=36. Since 36 is divisible by 9 so the number 890676 is also divisible by 9.

Divisibility by 10
If the last digit of the number is zero (0), then that number is fully divisible by 10.
Ex.:- 890 has 0 at the end, so it is divisible by 10.

Divisibility by 11
If the difference of the sum of its digits in odd places and the sum of its digits in even places is either zero (0) or a multiple of 11.
Ex.:- Let us consider the number 647053.
Sum of digits at odd places = 6 + 7 + 5= 18
Sum of digits at even places = 4 + 0 + 3= 7
Difference of the sums = 18 – 7=11
Since the difference 11 is a multiple of 11, so 647053 is divisible by 11.

Some General properties of Divisibility :-

1. If a number ‘x’ is divisible by another number ‘y’, then any number divisible by ‘x’, will also be divisible by ‘y’ and by all the factors of ‘y’.
Ex.:- The number 84 is divisible by 6. Thus any number that is divisible by 84, will also be divisible by 6 and also by the factors of 6, i.e. by 2 and by 3.

2. If a number ‘x’ is divisible by two or more than two co-prime number then x is also divisible by the product of those numbers.
Ex.:- The number 2520 is divisible by 5, 4, 13 that are prime to each other (i.e. co-prime), so, 2520 will also be divisible by 20 (= 5 x 4), 52 (= 4 x 13).

3. If two number ‘x’ and ‘y’ are divisible by a number ‘p’, then their sum x + y is also divisible by ‘p’.
Ex.:- The number 225 and 375 are both divisible by 5. Thus their sum = 225 + 375 = 600 will also be divisible by 5.
Note:- It is also true for more than two numbers.

4. If two number ‘x’ and ‘y’ are divisible by a number ‘p’, then their difference x – y is also divisible by ‘p’.
Ex.:- The number 126 and 507 are both divisible by 3. Thus their difference = 507 – 126 = 381 will also be divisible by 3.