LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

DIVISIBILITY - PROBLEM & SOLUTION

**DIVISIBILITY-PROBLEM & SOLUTION -**

**As we know so far the concept about divisibility, now we will learn the divisibility in more logical manner through problem solving.**

**Example.1) Write whether the following
numbers are divisible
by 6 or not. **

** Give reasons: a. 92,154 b. 84,020**

**Ans.)**

**DIVISIBILITY BY 6 :-**

**EXPLANATION :**

**Ø Make a table as shown in example**

**Ø First, check divisibility by 2- underline
the digit in the one’s place .**

**Ø If the underlined digit is even (0,
2, 4, 6, or 8), the given number is divisible**
**by 2.**

**Ø Next, check divisibility by 3 - find the sum of the digits of the given number .**

**Ø If the sum of the digits is divisible by 3, the given number
is divisible by 3.**

**Ø If the number is divisible both by 2 and 3, then the given number
is divisible by 6.**

**Ø But, if the number is divisible by any one (either 2 or 3) or none, then the given number is not divisible by 6.**

**Example.2) Write the smallest
number that should be (i) added to and (ii) subtracted from the following numbers to get them divisible
by 6.**

**Ans.) (a) 81**

**(i) +3**

**3 should be added (81+3=84)**

**(ii) -3**

**3 should be subtracted (81-3=78)**

**EXPLANATION -**

**[ Addition :-**

**It is not divisible
by 2 as the last digit is not even (1).**

**If we add 1, it will be 82, but 8+2=10 which is not divisible by 3. 84 is the next even number, which is divisible both by 2 and 3.]**

**[For Subtraction :-**

**80 (not divisible by 3) and 79 (not divisible by 2).**

**But, 78 is divisible
both by 2 and 3.]**

**Example.3) Write whether the following
numbers are divisible
by 9 or not. Give reasons:**

**a) 9,846, b) 53,103, c) 90,621**

**Example.4) Write the smallest number that should be (i) added to and (ii) subtracted from the following
numbers to get them divisible
by 9.**

**(b) 277**

**(i) +2**

**2 should be added (277+2=279)**

**(ii) -7**

**7 should be subtracted (277-7=270)**

**EXPLANATION :-**

**[ Addition :-**

**It is not divisible
by 9 as the sum of digits is 16 (2+7+7=16). We know 18 is divisible
by 9, we should add 2.**

**277+2 =279 is divisible by 9 (2+7+9=18). ]**

**[For Subtraction :-**

**2+7+7=16 is not divisible by 9.**

**16-7=9; 9 is
divisible by 9.**

**We should subtract 7; 277-7=270**

**2+7+0=9; 270 is divisible
by 9. ]**

**Example.5) Write whether the following numbers are divisible
by 11 or not. Give reasons: a. 86,284 b. 7,441**

**Ans.)**

**DIVISIBILITY BY 11 :-**

**EXPLANATION :-**

**Ø Make a table as shown in example**

**Ø
First, mark the odd and even places alternately in the table as shown.
The first place is always odd.**

**Ø
Find the sum of the digits that are in the odd places.**

**Ø Next, find the sum of the digits that are in the even places.**

**Ø Now, find the difference between these two sums (subtract).**

**Ø
If the difference is exactly divisible by 11, the given number
is divisible by 11.**

**Ø
Or, if the difference is zero (0), the given number is divisible
by 11.**

**Ø Otherwise the given number is not divisible
by 11.**

**We should find sums of the digits in both odd and even places and their difference) [O=ODD PLACE ; E= EVEN PLACE]**

**Example.6) Write whether
the following numbers are divisible
by 12 or not. Give reasons. a. 23,940 b. 78,618**

**Ans.) DIVISIBILITY BY 12 :-**

__EXPLANATION :-__

**Ø Make a table as shown in example**

**Ø First, check divisibility by 3 - find the sum of the digits
of the given number .**

**Ø If the sum of the digits is divisible by 3, the given number
is divisible by 3.**

**Ø Next, check divisibility by 4 - underline the digits in the ones and tens places (i.e. the last two digits).**

**Ø If the underlined last two digits can be divided by 4 leaving
no remainder, the given number is divisible
by 4.**

**Ø OR, if the underlined last
two digits are both zeroes (00), the given number is divisible by 4.**

**Ø If the number
is divisible both by 3 and 4, then the given number
is divisible by 12.**

**Ø But, if the number
is divisible by any one (either 3 or 4) or none,
then the given number is not divisible
by 12.**

**CHECK DIVISIBILITY BY 3 and 4 BOTH**