Informative line

Factorization

Factors

When we multiply two or more numbers to get another number (product) then the numbers which are being multiplied are known as factors of that another number.

For example:

  • If we have three numbers \(2,\;3\) and \(4\) then on multiplying these three numbers, we get

\(2×3×4=24\)

Here, \(2,\;3,\;\text{and}\;4\) are the factors of \(24\).

  • Now, if we multiply \(24\) with any number then \(24\) also becomes a factor.

For example:

\(24×3=72\)

Here, \(24\) and \(3\) are the factors of \(72\).

Factors in terms of divisibility rules

If a number \(a\) is divisible by \(b\) then \(b\) is a factor of \(a\).

For example:

Consider the number, \(30\).

We know that \(30\) is divisible by \(1,\;2,\;3,\;5,\;6,\;10,\;15,\;\text{and}\;30\).

Thus, all these are the factors of \(30\).

  • The factors of a number are always less than or equal to the number.

Illustration Questions

Which one is NOT a factor of \(16\)?

A \(2\)

B \(3\)

C \(4\)

D \(8\)

×

For option (A):

\(16\) is divisible by \(2\) because the last digit of 16 is even. 

So, 2 is a factor of \(16\).

For option (B):

The sum of all the digits of \(16 = 1 + 6 = 7\)  

But \(7\) is not a multiple of \(3\). So, \(16\) is not divisible by \(3\).

Thus, \(3\) is not a factor of \(16\).

For option (C):

\(16\) is a multiple of \(4\), so it is divisible by \(4\).

Thus, \(4\) is a factor of \(16\).

For option (D):

\(16\) is a multiple of \(8\), so it is divisible by \(8\).

Thus, \(8\) is a factor of \(16\).  

Hence, option (B) is correct.

Which one is NOT a factor of \(16\)?

A

\(2\)

.

B

\(3\)

C

\(4\)

D

\(8\)

Option B is Correct

Common Factors

  • The common factors always come into account when we factorize two or more numbers simultaneously.
  • In the list of the factors of the numbers, the factors that are common or same are called common factors.

For example:

We have two different numbers 16 and 18.

Factors of 16 = 2 × 8

Factors of 18 = 2 × 9

We observe that 2 is common in the list of the factors of both the numbers.

So, 2 is a common factor of 16 and 18.

Now consider three different number, 24, 36 and 60.

We observe that 2, 2 and 3 are common in the list of the factors of the numbers.

So, 2, 2 and 3 are the common factors of 24, 36 and 60.

Illustration Questions

Find the common factors of \(18\) and \(27\).

A \(3 × 3\)

B \(3 × 9\)

C \(2 × 9\)

D \(2 × 3\)

×

Factors of \(18 = 2 × 9 \)

\(= 2 × 3 × 3\)

Factors of \(27 = 3 × 9\)

\( = 3 × 3 × 3\)

\( 3\times 3\) are common in the list of the factors of both the numbers.

\(\therefore\) The common factors \(= 3 × 3\)

Hence, option (A) is correct.

Find the common factors of \(18\) and \(27\).

A

\(3 × 3\)

.

B

\(3 × 9\)

C

\(2 × 9\)

D

\(2 × 3\)

Option A is Correct

Prime Numbers and Composite Numbers

Prime numbers

  • A number which is divisible only by 1 and itself is called a prime number. The prime numbers are the whole numbers greater than 1.

For example: 2, 3, 5..... are divided evenly only by 1 and themselves, so they are all prime numbers.

  • To find divisibility of any number, check the divisibility with ascending order of prime numbers.

For example: 125

  • The digit at one's place is not zero or an even number, so it is not divisible by 2.
  • The sum of all digits ( 1 + 2 + 5 = 8 ) is not a multiple of 3, so it is not divisible by 3.
  • The digit at one's place is 5, so it is divisible by 5.

Composite Numbers

  • A number which is divisible by other numbers also, besides 1 and itself is called a composite number like 4, 6, 8, 9, 12,....

For example: 4 is divisible by 1, 2 and 4. Thus, other than 1 and 4, 2 is also a factor.

Hence, 4 is a composite number.

Illustration Questions

Which one of the following is correct?

A 33 is a composite and 11 is a prime number.

B 28 is a prime number and 41 is a composite number.

C 36 and 42, both are prime numbers.

D 31 and 29, both are composite numbers.

×

For the number 33:

33 is divisible by 3 and 11. So, it is a composite number.

For the number 11:

11 is divisible only by 1 and itself. So, it is a prime number.

Hence, option (A) is correct.

For the number 28:

28 is divisible by 2, 7 and 14. So, it is a composite number.

For the number 41:

41 is divisible only by 1 and itself. So, it is a prime number.

Hence, option (B) is incorrect.

For the number 36:

36 is divisible by 2, 3, 4, 6 and 9. So, it is a composite number.

For the number 42:

42 is divisible by 2, 3, 6, 7 and 21. So, it is a composite number.

Hence, option (C) is incorrect.

For the number 31:

31 is divisible only by 1 and itself. So, it is a prime number.

For the number 29:

29 is divisible only by 1 and itself. So, it is a prime number.

Hence, option (D) is incorrect.

Which one of the following is correct?

A

33 is a composite and 11 is a prime number.

.

B

28 is a prime number and 41 is a composite number.

C

36 and 42, both are prime numbers.

D

31 and 29, both are composite numbers.

Option A is Correct

Prime Factorization

  • The method of factorizing a number all the way down to the numbers which are prime (i.e. to the numbers that can not be further factorized) is called prime factorization and the numbers are called prime factors.

For example:

Prime factorization of \(12\)

\(=2×6\)

But \(6\) can be further factorized as-

\(2×3\)

So, \(12=2×2×3\) or \(2^2×3\), is prime factorization and \(2,\;2\) and \(3\) are the prime factors of \(12\).

  • We can easily understand it with the help of a factor tree.
  • A factor tree is a diagram of organizing factors and prime factors.

Prime factorization of \(36\) is shown through factor tree. 

So, prime factorization of \(36=2×2×3×3\) or \(2^2×3^2\) and \(2,\;2,\;3\) and \(3\) are the prime factors of \(36\).

Illustration Questions

Which one of the following represents the prime factorization of the given numbers?

A \(12=2×6\)

B \(18=2×3×3\)

C \(16=2×2×4\)

D \(20=4×5\)

×

The method of factorizing a number all the way down to the numbers which are prime (i.e. to the numbers that can not be further factorized) is called prime factorization and the numbers are called prime factors.

Option (A):

Prime factorization of \(12=2×2×3\)

image

Option (B):

Prime factorization of \(18=2×3×3\)

image

Option (C):

Prime factorization of \(16=2×2×2×2\)

image

Option (D):

Prime factorization of \(20=2×2×5\)

image

Hence, option (B) is correct.

Which one of the following represents the prime factorization of the given numbers?

A

\(12=2×6\)

.

B

\(18=2×3×3\)

C

\(16=2×2×4\)

D

\(20=4×5\)

Option B is Correct

Co-prime Numbers

  • The co-prime numbers are the set of two or more numbers which has only one (1) as a common factor. 
  • In other words, if there is no common number which divides the two different numbers then these two numbers are co-prime to each other.

For example: 14 and 15

Factors of 14 = 1 × 2 × 7

Factors of 15 = 1 × 3 × 5

There is only 1 as the common factor, that's why 14 and 15 are co-prime numbers. 

  • A single number is never a co-prime.
  • While dealing with co-prime we always deal with a set of numbers.
  • Two prime numbers are always co-prime to each other.

For example: 13 and 17

  • Two consecutive numbers are always co-prime because they don't have any common factor other than 1. 

For example: 15 and 16

Illustration Questions

Which pair is co-prime?

A \(13, 14\)

B \(6,8\)

C \(6,12\)

D \(12,14\)

×

The co-prime numbers are the set of two or more numbers which has only one (1) as a common factor. 

Option (A):

Factors of \(13 = 1 × 13\)

Factors of \(14 = 1 × 2 × 7\)

There is only \(1\), as the common factor.

Option (B):

Factors of \(6 = 2 × 3\)

Factors of \(8 = 2 × 2 × 2 \)

\(2\) is the common factor of \(6\) and \(8\).

Option (C):

Factors of \(6 = 2 × 3\)

Factors of \(12 = 2 × 2 × 3\)

\(2\)\(\times\) \(3\) are common factors of \(6\) and \(12\).

Option (D):

Factors of \(12 = 2 × 2 × 3\)

Factors of \(14 = 2 × 7 \)

\(2\) is the common factor of \(12\) and \(14\).

Hence, option (A) is correct.

Which pair is co-prime?

A

\(13, 14\)

.

B

\(6,8\)

C

\(6,12\)

D

\(12,14\)

Option A is Correct

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