Factors and Multiples

Factors and Multiples (عاد اور اضعاف)

Student learning outcomes:

Define a factor عادas a number which divides the dividend completely مکمل طور پرleaving zero (0) remainder.
Define a multiple اضعاف as a dividend into which a factor can divide.
Define even numbers as the numbers which are completely divisible by 2 or Multiples of two (2)
Define odd numbers طاق اعداد as the numbers which are not completely divisible by 2 or the numbers which are not Multiples of two (2).
Define prime numbers مفرد اعداد(2,3,5,7,11,13,17,.....), the numbers which have only two factors, 1st is 1 and other is itself.
Define composite numbers مرکب اعداد ( 4,6,8,9,10,12....), the numbers which have more than two factors.
Students are able to Know that 1 is neither prime nor composite as it has only one factor which is 1 itself.
Students are able to Know that 1 is a factor of every number.
Students are able to Know that 2 is only even prime number where as all other prime numbers are odd.
Test by inspection whether the numbers 2,3,4,5,6,8,9,11,12,15,25 can divide a given number completely.
Define prime factorization مفرد تجزی , the process of factorizing a number into its prime factors.
Recognize the index notationقوت نماٸ شکل.
Factorize تجزیa given number and express ظاہر کرنا its factors اجزا in the index notation(power form).
Define HCF (highest common factor) عاداعظمas the greatest number which is common factor of two or more numbers.
Find HCF( highest common factor) of two or more than two numbers by prime factorization and Long division method.
Define LCM ( least common multiple) ذواضعاف اقلas the smallest number which is a common multiple of two or more numbers.
Find LCM( least common multiple) of two or more than two numbers by prime factorization and Long division method.

Solve real life problems related to HCF (highest common factor) and LCM(least common multiple).

Factors:عاد

We know that if a number is divided ( تقسیم کیا جاے) by another number and the remainder (باقی) is zero (0) then the 1st number is said to be divisible (قابل تقسیم )by 2nd number.

For example:

From the above examples it can be seen that the number 18 is divisible by 1,2,3,6,9 and also by 18. These numbers are known as the factors (عاد) of the number 18, i.e, the factors of 18 are 1,2,3,6,9 and 18.

Similarly we can find the factors of any other numbers as giving

The factors of 12 = 1,2,3,4,6,12

The factors of 15 = 1,3,5,15

The factors of 4 = 1,2,3,6,7,14,21,42

So a factor of a number can be defined as(عاد کی تعریف)

A number that divides(تقسیم کرنا) the given number completely(پورا) is called a factor of the given number.

Note:

Every number has at least two factors(ہر نمبر کے کم از کم دو عاد ہوتے ہیں) except one (1)(سواے ایک کے). One has only 1 factor which is one (1) itself.

Multiples: اضعاف

All the numbers which are divisible (قابل تقسیم )by another number are called the multiples of that number.

For example:

Multiples of 2= 2,4,6,8,10,.......

Multiples of 3 = 3,6,9,12,.......

Multiples of 5 = 5,10,15,20,25,30,....

In the given example as we say that the factors of 18 are 1,2,3,6,9,18, we can also say that 18 is a multiple of each of the number 1,2,3,6,9,18.

The multiples of

2 = 2,4,6,8,10,12,14,16,18,20,....

The multiples of 3 = 3,6,9,12,15,18,21,....

The multiples of 9 = 9,18,27,36,.....

Thus a number is said to be a multiple(اضعاف) of each of its factor. It can also be noticed that the multiple of a number is either greater(بڑا) than or equal to the number itself.

Types of Natural numbers: قدرتی اعداد کی اقسام

Natural numbers are classified(جماعت بندی) in two ways, either even (جفت) and odd (طاق)or prime(مفرد) and composite(مرکب).

Even numbers: جفت اعداد

The numbers which are divisible (قابل تقسیم) by two (2) are called even Numbers. We can also say that all multiples of two (2) are even numbers. It mean 2,4,6,8,10,12.... are all even numbers because all numbers are multiple of two (2). The set of even Numbers is denoted(ظاہر کرنا) by captital letter (بڑے حروف)E.

E = { 2,4,6,8,......}

Odd numbers: طاق اعداد

The numbers which are not divisible by two (2) are called odd(طاق) numbers. It can also be said that the numbers which are not Multiples(اضعاف) of two (2) are odd numbers, i.e 1,3,5,7,9,11,... are all odd numbers. The set of odd numbers is denoted by the capital letter O.

O = { 1,3,5,7,....}

Prime numbers:مفرد اعداد

A number having exactly two factors(ایسے اعداد جن کے صرف دو عاد ہوں), one (1) and the number itself is called the prime number, i.e 2,3,5,7,11,13,17,.... are all prime numbers. The set of prime (مفرد) numbers is denoted(ظاہر کرنا) by the capital letter P.

P = { 2,3,5,7,11,.... }

Note:

The number two (2) is only even prime number where as all other prime numbers are odd.

Composite Number: مرکب اعداد

A number having factors other than two (2) (ایسے اعداد جن کے دو سے زیادہ عاد ہوں)and itself is called a composite Number or we can say that the numbers having more than two factors are composite numbers,i.e 4,6,8,9,10,12,... are all composite numbers because each of these numbers has more than two factors(عاد). The set of composite numbers(مرکب اعداد) is denoted (ظاہر کرنا) by the capital letter C.

C = { 4,6,8,9,10,12,....}

Note :

The numbers 1 is neither prime (نا ہی مفرد) nor composite (نا ہی مرکب) Number because it has only one factor which is 1 itself.

Examples:

Factors of 56 = 1,2,4,7,8,14,28,56

Factors of 121 = 1,11,121

Factors of 36 = 1,2,3,4,6,9,12,18,36

Multiples of 3 = 3,6,9,12,15,18,........

Multiples of 9 = 9,18,27,36,45,54,.......

Multiples of 12 = 12,24,36,48,60,........ e.t.c

Factorization: تجزی

The process (عمل) of writing a number into its factors(عاد) is called factorization.

We know that a natural number can be expressed as the product (حاصل ضرب) of its factors as giving below.

32 =( 2 )× (2) ×( 2) × (2)×(2)

42 = 2 × 3 × 7

So if a number which can be expressed as the product of prime factors (مفرد اجزاے ضربی) is called prime factorization.(مفرد تجزی)

The prime factors of a number can be written in any order like

42 = 2×3×7

42= 3×2×7 or

42 = 7×2×3

42 = 7×3×2

But often write them in ascending order(ترتیب صعودی).

The prime factors (مفرد عاد) of a number can be expressed by using a tree like diagram is called Factor Tree Method.

24 can be factorize by factor tree method

Index Notation: قوت نماٸ شکل

Look at the prime factors(مفرد عاد) of following numbers.

49 = 7×7

81 = 3×3×3×3

32 = 2×2×2×2×2

125 = 5×5×5

So we can write(لکھنا) the prime factors of above given Numbers as.

49 = 7×7 = 7² ( square of 7)

81 = 3×3×3×3 =3⁴ (3 to the power of 4)

32 = 2×2×2×2×2 =2⁵ (2 to the power of 5)

125 = 5×5×5 = 5³ ( 5 to the power of 3 or cube of 5)

Highest common factor: عاداعظم

To find HCF there are two methods

1- prime factorization method

2- Long division method

HCF by prime Factorization Method:

Lets consider the numbers 72, 48 and 132 and try to find their HCF by prime Factorization Method.

HCF by long Division Method:

Let's try to find HCF by long division method of the numbers 928 and 324.

The last divisor( آخری تقسیم کرنے والا) is the highest common divisor of two numbers.

So the HCF (عاداعظم) of 928 and 324 is 4.

Least common multiple (LCM): (ذواضعاف اقل)

The least common multiple ( LCM ذواضعاف اقل) of two or more than two numbers is the smallest (چھوٹا)number which is the multiple of each of the given numbers.

Look at the multiples of 4 and 6 which are given as

Multiples of

4 = 4,8,12,16,20,24,28,32,36,40,....

Multiples of 6 = 6,12,18,24,30,36,42,.......

We note that 12, 24, 36,.... Are the common Multiples of 4 and 6 but the smallest among (درمیان) then is 12, Hance 12 is known as the least common multiple of 4 and 6.

There are two methods to find the least common multiple (LCM).

1- Prime factorization method ( مفرد تجزی کا طریقہ)

2- Division Method (تقسیم کا طریقہ)

LCM by prime Factorization Method:

To understand this method we can solve an example. Lets find LCM of 36, 48 and 56 by prime Factorization method.

So LCM of 36,48 and 56 = 2⁴ × 3² ×7

LCM = 16 ×9 × 7

LCM = 1008

LCM by division method:

To find LCM by division method which is simpler than the prime factorization method. To understand this we solve the following example.

LCM of 24, 36, 54 and 81 by this method.