Divisibility rules - 7 and 11

Quote : Small minds discuss persons. Average minds discuss events. Great minds discuss ideas. Really great minds discuss mathematics.                                                                                  --Anon

How do you check a number is divisible by 7?

Pull out unit digit number, double and subtract the remaining number. Repeat this process till you get significant small number.

Let me take one example
Ex : 12896
step 1: pull out unit digit number i.e. 6
           double it i.e. 12
       2: 1289 - 12 = 1277
Repeat above process.
       127 - 2x7 = 127 - 14 = 113
 113 is not divisible by 7. So 12896 is not divisible by 7.

How do you check a number is divisible by 11?

This can be done by addition and subtraction. Add all odd place digits and even place numbers. If the difference of these two is 0 or divisible by 11, then the number is divisible by 11.

Let me take an example
Ex : 123454321
sum of odd place digits = 1 + 3 + 5 + 3 + 1 = 13
sum of even place digits= 2 + 4 + 4 + 2 = 12
difference of above sums = 13 - 12 = 1

It is not 0 or divisible by 11.
So 123454321 not divisible by 11.

So far from 2 to 11 divisibility rules covered.

Divisibility rules - 4 and 8

Quote : MATHEMATICS IS LIKE TRUE LOVE- A SIMPLE IDEA BUT CAN GET COMPLICATED... Ricky Gakhar

Friends,

How do you check a number is divisible by 4?

Take first two digits(units place and tens place) and divide with 4. If the two digit number is divisible by 4, the number is divisible by 4. 

Let me take one example

Ex1 : 2345672
step 1 : take first two digits. Here it is 72
step 2 : Is 72 divisible by 4?  Yes. So 2345672 is divisible by 4.

Find the number 56784345644 divisible by 4 or not?

How do you check a number is divisible by 8?
Take first three digits(units place, tens place and hundreds place) and divide with 8. If the three digit number is divisible by 8, the number is divisible by 8.

Let me take above example

Ex : 2345672
step 1 : take first three digits. Here it is 672
step 2 : Is 672 divisible by 8?  Yes. So 2345672 is divisible by 8.

Find the number 56784345644 divisible by 8 or not?

Divisibility rules - 3, 6 and 9

 Quote : If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.      

 -- Tobias Dantzig

How can you find a number is divisible by 3 or 6 or 9 ?

Very simple. Just find the sum of all digits in the number.

if sum is divisible by 3, the number is divisible by 3

if sum is divisible by 9, the number is divisible by 9

if sum is divisible by 3 and unit place digit is 0 or 2 or 4 or 6 or 8, the number is divisible by 6.

Lets look at examples.

a) 156
     step1 : sum = 1 + 5 + 6 = 12
     step2 : 12 is divisible by 3. So 156 is divisible by 3

b) 2345678
    sum = 2 + 3 + 4 + 5 + 6 + 7 + 8 = 35
    35 is not divisible by 3. So 2345678 is not divisible by 3
c) Is 135678 divisible by 9?
    sum = 1 + 3 + 5 + 6 + 7 + 8 = 30
    30 is not divisible by 9. So 135678 is not divisible by 9
d) Is 127 divisible by 6 ?
    unit place digit is 7. So 127 is not divisible by 6 ( Unit place digit should be 0 or 2 or 4 or 6 or 8 )

e) Is 1296 divisible 3 or 6 or 9 ?
    sum = 1 + 2 + 9 + 6 = 18

    18 is divisible by 3. So 1296 is divisible by 3

    18 is divisible by 9. So 1296 is divisible by 9

    Unit place digit is 6 and 18 is divisible by 3. So 1296 is divisible 6.

Check following numbers are divisible by 3 or 6 or 9
145678, 23567, 98741

Divisibility rules

Quote : MATHEMATICS is a great motivator for all humans.. Because its career starts with "ZERO" but it never end
 
Worked on sufficient multiplication examples and square of a number examples. Lets look into the division operation. In simplification every operation(addition or subtraction or multiplication and division) is very important.

How can you say whether a number "x" is divisible by a number "y"?

x can be any number and y is between 1 to 11 ( for simplicity purpose )




Lets look at divisibility rules.

How can you say whether a number is divisible by 2 or 5 or 10?

This is very simple. 
If unit place digit of the number is 0, 2, 4, 6 and 8, it can be divisible by 2.
If unit place digit of the number is 0 or 5, it can be divisible by 5.
If unit place digit of the number is 0, it can be divisible by 10. 

Ex : Is 1032 divisible by 2?
In the above example unit place digit is 2. So it is divisble by 2 and it is not divisible by 5 or 10.

Irrespective of magnitude of the number, just look at unit place digit to decide whether the number is divisible by 2 or 5 or 10.

Try following examples.
 1) 1025
 2) 10000000025
 3) 1111111111111111111111111111111111112
 4) 5555555555555555555555555550

Square of a number

Square of a number  by using base method:

I mentioned base method in two digit multiplication. So I will start with an example.

Ex 1 : 13 x 13

10 is the nearest round figure value. So 10 is base

13 = 10 + 3

13 ₊₃ x 13 ₊₃ = 13 +3 / 3 x3 = 16 / 9 = 169

Steps :

1. Find nearest round figure value

2. Follow base multiplication method

Ex 2: 14 x 14 = ?

14 x 14 = 14+4 / 4x4 = 18 / 16 = 196

Ex 3 : 23 x 23 = ?

Here base is 20

23 x 23 = 23+3 / 3x3 = 26 / 9

When base is 20, need to multiply 10's digit with 2

when base is 30, need to multiply 10's digit with 3

when base is 40, need to multiply 10's digit with 4

So

result = 26 x 2 / 9 = 52 / 9 = 529

Ex 4 : 44 x 44 = ?

Base = 40

result = 44 + 4 / 4x4 = 48 / 16 = 48 / 16
         = 48 x 4 /16 ( As base is 40 )
         = 192 / 16 ( 1 carry forward )
         = 1936

Try 56², 73² and 82²
 

Square of a number

Square of a number 

Square of a number can find in so many ways. But the important one is base method as it is multiplication with same number.

I will discuss here two methods i.e. 

1. Special case method

2. Base method

1. Special case method : 

    Rule : This method is applicable only when one's place digit is 5.

 

    As you know 5² = 25
    15² = 1x2 / 25 = 225

So steps are as follows.

Step 1 :  First two digits from right to left is 25

Step 2 :  Ten's digit x (Ten's digt + 1)

15 x 15 =  1x2 / 25 = 225
25 x 25 =  2x3 / 25 = 625
35 x 35 =  3x4 / 25 = 1225
45 x 45 =  4x5 / 25 = 2025
55 x 55 =  5x6 / 25 = 3025
65 x 65 =  6x7 / 25 = 4225
75 x 75 =  7x8 / 25 = 5625
85 x 85 =  8x9 / 25 = 7225
95 x 95 = ?

Very easy. Isn't it?

Next We will look into base method

Two digit multiplication with 50

Ex 1 : 18 x 50 = ?

Multiplication with 50 may take time for you. But if I ask
50 paise coins 18 = How many rupees

You will immediately tells 9 rupees. It is nothing but 18 x 50 = 900

When you are multiplying with 25 or 50, just think in paise. You will get answer quickly.

Lets how to do it in easy way. To do multiplication with 50, you need to remember table 2.

For example 18 x 50 = ?

when you divide 18 with 2, result is 9 and remainder is 0

So result is 900

Let me take one more example

Ex 2: 17 x 50 =  ?

When you divide 17 with 2, result is 8 and remainder is 1
So result is 850

Please note if remainder is 0, last two digits in the result  00

                   if remainder is 1, last two digits in the result 50  

Let me take more examples

Ex 3: 24 x 50 = 1200 (24 divided 2  is 12 and remainder 0)
Ex 4: 35 x 50 = 1750 (35 divided 2  is 17 and remainder 1)

Start practicing two digit multiplication...