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