## Arithmetic Chapter 10 Highest Common Factor And Least Common Multiple Or 3 Numbers

**Arithmetic Chapter 10** **Factor and Multiple**

- In the operation of division, a number is divided by another number.
- A number that is divided is called the
and the number by which division is performed is called the**Dividend****Divisor.** - The result of the operation of division is called the
**Quotient.** - The consecutive positive integers starting from 1 i.e., the numbers 1, 2, 3, 4, … . are called
**Natural Numbers.** by the second number.**divisible**- Here the first number is called the dividend and the second number is called the divisor.
- The result of the division is called the quotient.
- For example, when 27 is divided by 3, then there will be no remainder and so we say that 27 is divisible by 3.
- Here 27 is the dividend, 3 is the divisor and the result of the division is 9 which is the quotient.
- If a number is divisible by another, then the dividend is called the
of the divisor and the divisor is called a**Multiple**of the dividend. In the above example, 27 is a multiple of 3, and 3 is a factor of 27.**Factor** - Similarly, if 65 is divided by 13, then there will be no remainder and so 65 is divisible by 13. Here 65 is a multiple of 13 and 13 is a factor of 65.
- The natural numbers which are divisible by 2 are called
and which are not divisible by 2 are called**Even numbers**2, 4, 6, 8, etc are even numbers and 1, 3, 5, 7, etc. are odd numbers.**Odd Numbers.**

## Arithmetic Chapter 10 Common Factor

- A number that is a factor of two or more numbers is called a
of those numbers.**Common Factor** - For example : 15 = 3 x 5; 25 = 5 x 5; 35 = 5 x 7.
- Here we see that 5 is a factor of 3 numbers 15, 25, and 35.
- So 5 is a common factor of the numbers 15, 25, and 35.

## Arithmetic Chapter 10 Greatest Or Highest Common Factor

- A composite number has two or more factors. The highest (or greatest) among all possible common factors of two or more numbers is called the
of those numbers and abbreviated as**Highest (or Greatest) Common Factor**or**H.C.F.****G.C.F.** - For example, we consider the numbers 36 and 48.
**∴ 36 has factors:**1, 2, 3, 4, 6, 9, 12, 18, 36;**∴ 48 has factors:**1, 2, 4, 6, 8, 12, 16, 24, and 48.**36 and 48 have common factors:**1, 2, 6, and 12.- Among these factors, the highest common factor is 12. So, H.C.F. of 36 and 48 is 12.

## Arithmetic Chapter 10 Determination Of Highest Common Factor Of Three Numbers

- In order to determine the H.C.F. of three or more numbers, first we find the
of the numbers.**prime factors** - Then choose the common factors of the numbers. The product of these common prime factors is the required H.C.F. of the numbers.
- For example 84 = 2 x 2 x 3 x 7

126 = 2 x 3 x 3 x 7

210 = 2 x 3 x 5 x 7 - 2, 3, and 7 are the common factors of 84, 126, and 210.
- ∴ The H.C.F. = 2 x 3 x 7 = 42.

## Arithmetic Chapter 10 Highest Common Factor Of Compound Quantities

- In order to obtain the H.C.F. of two or more compound quantities, they are expressed in the lowest- order of units. Then determine the H.C.F. of them and put the unit.
- Observe the following example

**Example: ****Find the H.C.F. of 8 kg 981 gm; 14 kg 113 gm and 16 kg 679 gm. **

** Solution: **8 kg 981 gm = 8981 gms.

14 kg 113 gm = 14113 gms.

16 kg 679 gm = 16679 gms.

First, find the H.C.F. of 8981 gms and 14113 gms

H.C.F. of 8981 gms and 14113 gms = 1283 gms.

Now we shall find the H.C.F. of 1283 gms and 16679 gms.

H.C.F of 1283 gms and 16679 gms = 1283 gms.

H.C.F. of 8981 gms., 14113 gms., 16679 gms. = 1283 gms.

So the required H.C.F. = 1283 gms. = 1 kg = 283 gm.

## Arithmetic Chapter 10 Common Multiple or Least (Lowest) Common Multiple

- There are many multiples of a natural number.
- Two or more natural numbers may have an infinite number of common multiples.
- The last (or lowest) common multiple of all the common multiples of two or more natural numbers is called the
**L.C.M. of the Least (or lowest) Common Multiple.** - For example, we consider the numbers 12, 15, 20, and 30.
**Multiple of 12 is**12, 24, 36, 48, 60, 72, 84, 96, 108, 120, etc.**Multiple of 15 is**15, 30, 45, 60, 75, 90, 105, 120, etc.**Multiple of 20 is**20, 40, 60, 80, 100, 120, 140, 160, etc.**Multiple of 30 is**30, 60, 90, 120, 150, 180, 210, 240, etc.- From above we see that the common multiple of 12, 15, 20, and 30 are 60, 120, 18.0, etc.
- Among them, 60 is the smallest common multiple.
- ∴ The lowest common multiple of 12, 15, 20, and 30 = 60, and hence L.C.M. of then = 60.
- It is clear that the L.C.M. of two or more given numbers must be divisible by the numbers.

## Arithmetic Chapter 10 Determine Of Least Common Multiple Of Three Numbers

- In order to determine the L.C.M. of three or more given numbers, first of all, find the prime factors of the numbers individually and then determine the common prime factors of the numbers.
- Next, take other factors (other than the common factors) remaining of all the numbers and omit the factor or factors which have already been taken.
- The product of these factors and the common prime factors will give the L.C.M. of the numbers.
- For example, we take the numbers 12, 16, and 20.

12= 2 x 2 x 3;

16= 2 x 2 x 2 x 2;

20= 2 x 2 x 5. - The common prime factors = 2, 2, and the other factors remaining in the numbers are 3; 2, 2; 5 respectively.
- Therefore the L.C.M. of 12, 16 and

20 = 2 x 2 x 3 x 2 x 2 x 5

= 240.

## Arithmetic Chapter 10 Determination of Least Common Multiple of compound quantities

- In order to determine the L.C.M. of two or more compound quantities, the quantities should be expressed in the lowest order or units.
- Then obtain the L.C.M. of them in the usual process and put the unit to this L.C.M.
- This gives the required L.C.M.
- Observe the following example

**Example:** **Find the L.C.M. of 30 min. 48 sec.; 46 min. 12 sec. and 1hr. 17 min.**

** Solution**: 30 min. 48 sec. = 1848 sec.

46 min. 12 sec. = 2772 sec.

1 hour. 17 min. = 4620 sec.

L.C.M. of 1848, 2772, 4620 = 2 x 2 x 3 x 11 x 7 x 2 x 3 x 5 = 27720

The required L.C.M. = 27720 sec.

= 462 min.

= 7 hr. 42 min.

The required L.C.M = 7 hr. 42 min.

## Arithmetic Chapter 10 Relation between Highest Common Factor and Least Common Multiple of two numbers

- Let 22 and 33 be two given numbers.
- Then H.C.F. of 22 and 33 = 11; L.C.M. of 22 and 33 = 66
**.** - H.C.F. x L.C.M. = 11 x 66

=11 x 2 x 33

= 22 x 33

= Product of two given numbers. - ∴ Product of two given numbers = Their H.C.F. x L.C.M.
- ∴ L.C.M. = Product of two numbers ÷ H.C.F.
- H.C.F. = Product of two numbers ÷ L.C.M.
- Thus dividing the product of two numbers by their H.C.F. we can find the L.C.M. of the numbers.
- Again dividing the product of two numbers by their L.C.M., we can find H.C.F.
- This formula is not true for three (3) numbers.