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Prime factorization is a method of making an original number by multiplying two prime numbers. The process of factoring a number with respect to prime numbers is known as prime factorization. The prime factor of a number is calculated so that the original number is easily divisible by the factors. The division method and the factor tree method are the two commonly used prime factorization methods. The first few prime numbers are 2, 3,5,7, 11,13,17, and so on.
Read Also: Area of a Triangle
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Key terms: Prime Factorization, Prime Factors, Prime Numbers, Composite Number, Method, Real Numbers, Original Numbers, Whole Numbers
Prime factorization of HCF, and LCM
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A composite number is produced when a prime number is multiplied by any natural or whole number. Prime factors are found by factoring composite numbers by the method of prime factorization. The HCF and LCM of a set of numbers can also be calculated with the help of prime factorization.
HCF and LCM of a number can be found by the division method and prime factorization method. The product of several of the factors of a number is known as factorization. Factorization of HCF and LCM is used to find the smallest constituent of a number and to find out the number as the product of the factors.
Check Important Relation between HCF and LCM
Prime Factors of a Number
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A real number can be found by multiplying two prime numbers together. There is a complete division of the original number by the prime number. It is equivalent to factoring a number and only the prime numbers are taken into consideration among the factors. Eg, the prime factors of 10 are 2 and 5.
when the original number is completely divided by the factors and cannot be divided into more factors then it is known as prime factors of a number. some of the examples are :
42 = 2 x 3 x 7
30 = 2 x 3 x 5
48 = 2 x 2 x 2 x 2 x 3
Different Methods Of Prime Factorization
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There are two methods of prime factorization:
- Division method: the method of finding the factors of a large number is the ideal step to calculate the factors of a prime number. the original number should be divided completely by the smallest prime number. The resulting quotient should be divided by the small prime number until the quotient becomes 1. Lastly, multiply all the prime factors.
- Factor tree method: note the pair of factors of a given number. factorize the composite factors and write down the factor pairs as branches. Continue factoring until all the prime factors of the composite numbers are found.
Check Also: Euclid Division Lemma
Things to Remember
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- The prime factor of a number can be found by dividing the original numbers by prime factors until the result is one. A prime number cannot be factorized as it can only be divided by 1 or itself.
- Composite numbers are produced with prime numbers that are multiplied by any natural number.
- The method of prime factorization applies only to composite numbers because composite numbers have more than two factors.
- In the factor tree, factorization of the composite numbers continues until there is a closure.
- Breaking of secret codes based on numbers which are also known as cryptography is possible through prime factorization.
- Factors are numbers that can be multiplied together to get another number. they can be algebraic variables, numbers, or algebraic expressions.
Check Important Formulas for Logarithm
Sample Questions
Ques. Find the prime factor of 12. (2 marks)
Ans. 12 ÷ 2 = 6
Because 6 is not a prime number,
6 ÷ 2 = 3
3 is a prime number, thus
12 = 2 x 2 x 3
Or
12 = 22 x 3
Ques. What is Prime Factorization of 147? (3 marks)
Ans. 147 ÷ 3 = 49
147 is not divisible by 2 as it does not give a whole number.
49 ÷ 7 = 7
As the next prime number, 5 does not give a whole number
147 = 3 x 7 x 7
Or
147 = 3 x 72
Ques. Factorize 12a2b + 15ab2 (2 Marks)
Ans. 12a2b = 2 x 2x3 x a x a x b
15ab2 = 3 x 5 x a x b x b
The common factors in both the equations are 3, a and b
12a2b + 15ab2 = (3 x a x b x 2 x 2 x a ) + (3 x a x b x 5 x b )
= 3 x a x b x [(2 x 2 x a) +(5 x b )]
= 3 ab x (4a + 5b)
= 3 ab (4a + 5b)
Ques. Factorize 6xy – 4y +6 – 9x. (5 marks)
Ans. There is no common factor among all terms, although the first two terms have a common factor 2y.
6xy – 4y = 2y (3x -2 ) … (a)
-9x +6 = -3(3x)+3 (2)
= -3 (3x- 2) …. (b)
Solving (a) and (b) together
6xy -4y + 6 – 9x = 6xy – 4y – 9x + 6
= 2y(3x-2) – 3(3x-2)
= (3x-2) (2y-3)
Thus, the factors of 6xy – 4y +6 – 9x are (3x-2) and (2y-3)
Ques. Factorize m4 – 256. (3 marks)
Ans. m4 = (m2)2 and 256 = (16)2
(m2 -16)(m2+16)
(m2 -16) = m2 -42
=(m-4)(m+4)
Thus, m4 – 256 = (m-4)(m+4) (m2+ 16)
Ques. Find the factors of 3m2+ 9m+ 6. (2 marks)
Ans. 3m2 + 9m + 6 = 3(m2 + 3m + 2)
m2 + 3m+ 2 = m2 +m+ 2m + 2
= m(m+1) +2(m+1)
=(m+1)(m+2)
Therefore, 3m2 + 9m + 6 = 3(m+1)(m+2)
Ques. Determine Prime Factorization of the following number:
(a)28 , (b) 100, (c)32, (d) 900 (e) 126 (5 marks)
Ans.
- 28 = 2 x 2 x 7 = 22 x 7
- 100 = 2 x 2 x 5 x 5 = 22 x 52
- 32 = 2 x 2 x 2 x 2 x 2 = 25
- 900 = 2 x 2 x 3 x 3 x 5 x 5 = 223252
- 126 = 2 x 2 x 3 x 3 x 7 = 22327
Ques. Find Prime Factorization of A x B if A = 23 x 52x 11 and B = 22 x 5 x 13. (2 marks)
Ans. A x B = (23 x 52 x 11) x (22 x 5 x 13)
= 2 x 2 x 2 x 5 x 5 x 11 x 2 x 2 x 5 x 13
=25 x 53 x 11 x13
Ques. Find Prime Factorization of 70 and 100. Also find Prime Factorization of 7000 when 7000=100 x 70. (3 marks)
Ans. 100 = 22 x 52
70 = 2 x 5 x 7
Prime factorization of 7000 when 7000= 100 x 70
7000 = 100 x 70
= (22 x 52 ) x (2 x 5 x 7)
= 2 x 2 x 5 x 5 x 2 x 5 x 7
=23 x 53 x 7
Ques. with the help of prime factorization method, find the HCF of 75 and 50 (2 marks)
Ans. Prime factors of 75 = 3 x 5 x 5
prime factors of 50 = 2x5 x 5
the common factors are= 5 x 5
HCF of (75,50) = 25
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