We will discuss here about the concept of exact divisibility.

When we divide 21 by 3, it leaves no remainder. We say that 21 is exactly divisible by 3. Again, it we divide 21 by 4, it leaves 1 as remainder. We say that 21 is not divisible by 4.

A number is exactly divisible by another number only when it leaves 0 as remainder when we divide it.

But to find whether a number is exactly divisible or not, we apply certain divisibility tests. These tests help us to ascertain the exact divisibility of a number.

**Exact Divisibility by 2**

2 divide all even numbers exactly such as 2, 4, 6, 8, 12, 14, 16, 18, 20, etc. We see that the unit digit of these numbers is 0, 2, 4, 6 or 8.

The product of 2 and a whole number is called an even number.

A number is exactly divisible by 2 it its unit digit is 0,
2, 4 6 or 8.

Example:

Find whether the numbers 120, 232, 354, 456, 568 are exactly divisible by 2 or not.

Solution: 120, 232, 354, 456, 568 are exactly divisible by 2, as their last digits are 0, 2, 4, 6 and 8 respectively.

**Exact Divisibility by 3**

Let us divide 24 by 3.

24 ÷ 3 = 8. Here, 24 is exactly divisible by 3.

We can check the divisibility of any number by 3 with a test. That is – If the sum of the digits of a number is exactly divisible by 3, the number is exactly divisible by 3.

As in case of number 24. See the sum of digits of 24.

2 + 4 = 6 which is exactly divisible by 3 (6 ÷ 3 = 2)

Consider some more numbers.

12 → 1 + 2 = 3

18 → 1 + 8 = 9

21 → 2 + 1 = 3

24 → 2 + 4 = 6

27 → 2 + 7 = 9

30 → 3 + 0 = 3

Here, the sum of the digits of each number is exactly divisible by 3. Hence, these numbers are exactly divisible by 3.

Example: Find whether 1350 is exactly divisible by 3 or not.

Solution: Add the digits of 1350.

1 + 3 + 5 + 0 = 9, which is exactly divisible by 3 (9 ÷ 3 = 3)

Hence, 1350 is exactly divisible by 3.

**Exact Divisibility by 4**

100, 200, 300, 400, ...., etc are all exactly divisible by 4. So, we say that all whole hundreds are exactly divisible by 4. Now, consider the number 456. We can write it as 456 = 400 + 56. We know, 400 is exactly divisible by 4. 56 is also exactly divisible by 4. Thus, 456 is exactly divisible by 4. Exactly divisible by 4 of a n umber depends on the last two digits of the number. Observe the number 456. Its last two digits from a new number 56. Now, 56 ÷ 4 = 14

It means 56 is exactly divisible by 4. So, 456 is also exactly divisible by 4.

If the number formed by the last two digits of a number is exactly divisible by 4, the number is exactly divisible by 4.

Example Find whether 9348 is exactly divisible by 4 or not.

Solution: The number formed with the last two digits of the given number, is 48 which is exactly divisible by 4 (48 ÷ 4 = 12). Hence, the number 9248 is exactly divisible by 4.

**Exact Divisibility by 5**

Observe the multiples of 5.

5 × 1 = 5 5 × 2 = 10 5 × 3 = 15
5 × 4 = 20 5 × 5 = 25 |
5 × 6 = 30 5 × 7 = 35
5 × 8 = 40 5 × 9 = 45 5 × 10 = 50 |

You see that the multiples of 5 have either 5 or 0 as last digit.

(A multiple of a number is the product of the number and another whole number.)

A number is exactly divisible by 5 if its last digit is 0 or 5.

Example: Find whether 765 and 960 are exactly divisible by 5 or not.

Solution:

765 has 5 as its unit digit. So, it is exactly divisible by 5.

960 has 0 as its unit digit. So, it is exactly divisible by 5.

**Exact Divisibility by 6**

We know that 6 = 2 × 3

A number is exactly divisible by 6, if it is exactly divisible by 2 as well as 3.

For example,

Find whether 7512 and 4368 are exactly divisible by 6 or not.

7512 is exactly divisible by 2 as well as 3 (7512 ÷ 2 = 3756 and 7512 ÷ 3 = 2504). So, 7512 is exactly divisible by 6.

Similarly, 4368 is exactly divisible by 2 as well as 3 (4368 ÷ 2 = 2184 and 4368 ÷ 3 = 1456). So, 4368 is also exactly divisible by 6.

**Exact Divisibility by 7**

A number is exactly divisible by 7 if the difference between twice of the last digit and the number formed by remaining digits is either 0 or a multiple of 7.

For example,

Find whether 882 is exactly divisible by 7 or not.

Last digit of 882 is 2. Twice of 2 is 4. Now, 88 - 4 = 84 which is exactly divisible by 7 (84 ÷ 7 = 12). Hence, 882 is exactly divisible by 7.

Find whether 994 is exactly divisible by 7 or not.

Last digit of 994 is 4. Twice of 4 is 8. Now, 99 - 8 = 91 which is exactly divisible by 7 (91 ÷ 7 = 13). Hence, 994 is also exactly divisible by 7.

**Exact Divisibility by 8**

A number is exactly divisible by 8 if the number formed by its last three digits is exactly divisible by 8.

For example,

Find whether 4323 is exactly divisible by 8 or not.

The number formed by the last three digits of the number 4323 is 323 which is not exactly divisible by 8 (323 ÷ 8 = 40 and remainder 3). Hence, 4323 is not exactly divisible by 8.

**Exact Divisibility by 9**

If the sum of the digits of a number is exactly divisible by 9, the number is also exactly divisible by 9.

For example,

18 → 1 + 8 = 9

54 → 5 + 4 = 9

27 → 2 + 7 = 9

576 → 5 + 7 + 6 = 18

36 → 3 + 6 = 9

936 → 9 + 3 + 6 = 18

We see, the sum of the digits of each number is exactly divisible by 9. Hence, these numbers are also exactly divisible by 9.

For example,

Find whether 9199 is exactly divisible by 9 or not.

The sum of the digits of 9199 is 9 + 1 + 9 + 9 = 28 which is not exactly divisible by 9 (28 ÷ 9 = 3 and remainder 1). Hence, the number 9199 is not exactly divisible by 9.

**Exact Divisibility by 10**

Observe the multiples of 10.

10 × 1 = 10
10 × 2 = 20 10 × 3 = 30 10 × 4 = 40 10 × 5 = 50 |
10 × 6 = 60
10 × 7 = 70 10 × 8 = 80 10 × 9 = 90 10 × 10 = 100 |

We find that the unit digit of multiples of 10 is 0.

A number having 0 as unit digit is exactly divisible by 10.

For example,

Find whether 1980 is exactly divisible by 10 or not.

The unit digit of 1980 is 0. Hence, 1980 is exactly divisible by 10. Numbers like 0, 30, 50, 80, 100, 120, 180, 360, 960, 1280, etc are all exactly divisible by 10 because their unit digit is 0.

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