LCM - Least Common Multiple

Quote: Mathematics is the door and key to the sciences.   ---Roger Bacon

Ex1. Find LCM 2 and 4?

Multiples of 2 : 2, 4, 6, 8,...
Multiples of 4 : 4, 8, 12, 16,...
Common multiples of 2 and 4 : 4, 8, ...
Least common multiple : 4

Rules from above example:

1. If one number is multiple of other number, the bigger number is LCM

2. To find LCM easily take the small number multiples and check multiple is divisible by bigger number. If yes, then the multiple is LCM

Ex2 : find LCM of 3 and 9?

LCM is 9 as 9 is divisible by 3.

Ex3. Find LCM of 4 and 6?

Small number is 4
Multiples of 4 : 4, 8, 12, 16, ...
Is 4 divisible by 6? No
Is 8 divisible by 6? No
Is 12 divisible by 6? Yes. So LCM is12

Ex4. Find LCM of 3 and 4?

multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
So LCM is 12

Rule: if both the numbers doesn't have common factors, then LCM is product of two numbers.

Lets see how to find LCM  for more than two numbers in next post

Comparision of fractions

Quote: The essence of mathematics is not to make simple things complicated, but to make complicated things simple.    --S.Gudder

Let me take one example
1. compare 1/4 and 3/4 
3/4 > 1/4

Rule 1 : If denominators of the both fractions are equal, compare numerators.

In the above example denominator of the both fractions is 4.
So compare 1 and 3.
As 3 > 1, 3/4 > 1/4

2. compare 3/7 and 3/9
 3/7 > 3/9

Rule 2 : if numerators of the both fractions are equal, denominator with more value is lesser.

 In the above example 3/7 and 3/9 having same numerator. So denominator with more value is lesser.
 In this case denominator 9 is more. So 3/9 is lesser than 3/7

3. compare 2/3 and 4/5 
    step 1 : 2/3   ?  4/5
    step 2 : 2x2/3x2 ? 4/5 ( multiply denominator and numerator with same value i.e. 2)
    Multiplying with 2 to make both the numerator will become same.

4. compare  2/7 and 5/9

    This is generic method and can be applied every example above. 

    Known as Cross multiplication method.

    step 1 : 2/7 , 5/9  ( cross multiplication : multiply numerator and denominator )

                2x9  ?  5x7

                18    ?  35

                18    <  35

          So  2/7 < 5/9

    In traditional method, need to find Least Common Multiple(L C M).
    LCM of 9 and 7 is 63 (9 x 7) as both are prime numbers. 

    So 2/7 x 9/9 and 5/9x7/7
          18/63 ? 35/63
          As denominators are equal 18/63 < 35/63 
          Hence 2/7 < 5/9

Lets look how to find LCM in next post...
                                                                                                                                         

Fractions -- Introduction

Quote : Do not worry about your difficulties in mathematics, I aessure you that mine are greater.
                                                                              --Albert Einstein  

What is fraction?

Dividing into equal pieces or parts.

When  fractions are used?

Let me take one example. Two friends want to share 10 rupees. how much each one of them get?

Answer is 5. But how did you get that. By dividing 10 into 2 parts. So by division whole thing divided into equal parts

In the above example each part is Rs. 5/- i.e 10 x 1 / 2

In 1/2
1 is numerator
2 is denominator

Let me ask you one question.
If the two friends divide Rs. 10/- into 4 parts and each one takes two parts. Then each one will get how many rupees? Then also Rs.5/- only

So 10 x1/2 or 10 x 2/4 or 10 x 3/6..... all are equal.

Rule : if you multiply numerator and denominator with the same number, there is no change in the value of fraction.

1/2 = 2/4 = 3/6

Fractions can be expressed as decimal values. For example
1/2 = 0.5
1/4= 0.25


In the above example whole amount i.e. Rs. 10/-
how many friends sharing i.e. denominator


if 3 friends divide Rs. 10/-  then one part value is 10 x 1 / 3
if 4 friends divide Rs. 10/-  then one part value is 10 x 1 / 4
if 5 friends divide Rs. 10/-  then one part value is 10  x 1 / 5

each part value = whole value / no. of friends sharing

Human computer shakuntala Devi

Shakuntala Devi, Great mathematician by birth, died on Sunday morning. Universities recognised her talent in her childhood days. We may not have such a great mathematician again.

About her : Shakuntala Devi

Some excerpts from wiki pedia about her achievements.

In 1977 in Dallas she competed with a computer to see who could give the cube root of 188132517 faster. She won. At an American university she was asked to give the 23rd root of 91674867692003915809866092758538016248310668014430862240712651642793465704086709659 3279205767480806790022783016354924852380335745316935111903596577547340075681688305620821016129132845564805780158806771. She answered in 50 seconds. Her answer of 546372891 took a UNIVAC 1108 computer a full minute (10 seconds more) to confirm that she was right after it was fed with 13000 instructions. [2]

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...

Two digit multiplication with 25

Two digit Multiplication with 25

When you are multiplying with 25, you need to remember table 4 and table 25.

Table 4 is for division and Table 25 for multiplication.

Just you need to remember
25 x 1 = 25
25 x 2 = 50
25 x 3 = 75

Let me take one example

Ex 1. 13 x 25 = ?

Take 13 and divide with 4.

There are 3 fours in 13 and remainder is 1

result = 3x100 + 25 x 1 = 300 + 25 = 325

So you need to find how many 4s are there in the number and what is remainder.

Let us try one more example

Ex 2 : 22 x 25 = ?

5 fours are there in 22 and remainder is 2

result = 500 + 2 x 25 = 550

Rule is simple

step 1 : find remainder and how many fours are there in the number

step 2 : no. of 4s x 100 + remainder x 25

Let me take two more examples

Ex 3 : 35 x 25 = 8 x 100 + 3 x 25 = 875
Ex 4 : 48 x 25 = 12 x 100 + 0 x 25 = 1200

conclusion :

if remainder is 0 last two digits 00

if remainder is 1 last two digits 25

if remainder is 2 last two digits 50

if remainder is 3 last two digits 75

Last example

Ex 5: 81 x 25 =  2025
Similar way we can do for 50. Lets see this in next post

Two digit multiplication with 99

Today we will try multiplication with 99.

Ex 1 : 23 x 99

Multiplication with 99 can be done by subtraction

result = 23 -1 / 99 - (23 -1 )
          = 22 /  99-22
          = 22 / 77
          = 2277

How to do?

1. Just subtract 1 from number(not 99). So here other than 99 is 23 

Hence 23 - 1 = 22

2. Subtract resultant value from 99

 Hence 99 - 22 =  77

Result = 22 / 77 = 2277

Please note here also there is no carry forward

Try following examples

56 x 99
34 x 99
76 x 99
85 x 99

Lets see how to multiply with 25 in next post