Interesting Number Theory In Math

NUMBER THEORY

HCF AND LCM

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FACTORS AND MULTIPLES

When a number ‘a’ is divided with another number ‘b’

exactly leaving no remainder. There b is factor of a , or

alternatively I can say a is multiple of b. A common factor

is a number which divide them exactly leaving no

remainder.

For example, 2 is factor of 6 and 6 is multiple of 2.


LCM(LEAST COMMON MULTIPLE)

LCM is the smallest non zero number that is common in two

or

more numbers. It is the smallest number that is multiples of

numbers.

For example:

Multiples of 8 are: 8,16,24,32,……………

Multiples of 24 are: 24,48,72,96,……………..

Therefore, the smallest common multiples between 8 and 24

is 24.


METHODS TO FIND THE LCM

Following are the methods to find the LCM

Factorization

Division


FACTORIZATION

In factorization, we will break or factorize the number in prime factors and

them multiply all the prime factors common to all the numbers.

For example:

LCM of 12, 15, 20 is 60.

Factors of 12 are: 2x2x3

Factors of 15 are: 3x5

Factors of 20 are: 2x2x5

So, factors present in all the numbers are: 2x2x3x5=60

Therefore, LCM of 12, 15 and 20 is 60.


DIVISION METHODIn division method, we divide numbers to their prime factors.

For example: LCM of 12, 27 and 40


HIGHEST COMMON FACTOR (HCF)

The highest common factor is the largest factor that two

numbers have in common.

Let’s find the GCF of 12 and 42. First, we need to make a

list of factors for each number.

This is also called as greatest common factor (GCF)


METHODS TO FIND THE HCF

Following are the methods to find the HCF

Factorization

Division


USING PRIME FACTORIZATION

We need to find the prime factors of each

number and then Identify the common

factors. Then we multiply them together.


FACTORIZATION METHODEXAMPLE: HCF OF 27 AND 36

27

3 x 9

3 x 3

36

2 x 18

2 x 9

3 x 3

•Both prime factorizations

have 3x3 in common.

•Therefore, the HCF is 9.


DIVISION METHOD

HCF is the highest common number that is

present in two or more numbers. We can also

find HCF by upside down division.

Example: To find the HCF of 12 and 42


12

1 x 12

2 x 6

3 x 4

4 x 3

Factors of 12:

1, 2, 3, 4, 6,12

42

1 x 42

2 x 21

3 x 14

4 x ??

5 x ??

6 x 7

7 x 6

Factors of 42:

1, 2, 3, 6, 7, 14, 21,

42

Common Factors: 1, 2, 3, 6

Greatest Common Factor: 6 12© 2 0 1 7 V N A Y A . C O M I N C . A L L R I G H T S R E S E R V E D

WORD PROBLEMS ON LCM AND HCF

1. Judy has 16 red balloons and 24 yellow balloons. She wants to give same

number of each color balloons to kids. What is the greatest number of

Kids she can give balloons?

Answer:

The greatest common number is : 8

So, she can give balloons to 8 kids with same number of red and yellow.


2. Two traffic lights are turned on at the same time. one light turned on every 9

seconds. the other turned on every 15 seconds. after how many seconds will they on

together again?

Answer:

For the first light: 0, 9, 18, 27, 36, 45, 54, 63,

And for the second light: 0, 15, 30, 45, 60, 75

Therefore , the smallest common number is : 45

So they will turned again after 45 seconds.


3. Jerusha has 12 oranges, 28 apples, and 36 bananas. what is the greatest number

of baskets she can make if each type of fruit is equally distributed among the

baskets? how many of each supply will go into the baskets?

Answer:

4 baskets with 3 oranges, 7apples, and 9 bananas.


4.Two clocks are turned on at the same time. one clock chimes every 15 minutes.

the other clock chimes every 25 minutes. in how many minutes will they chime

together?

Answer:

First clock: 0,15,30,45,60,75,90,105,,,,,,,,

And second clock: 0,25,50,75,100,125,,,,,,,

So, smallest common number is : 75


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