Informative line

Least Common Multiple (LCM) of Prime and Co-prime Numbers

LCM of prime numbers

  • Since the prime numbers do not have any common factor other than 1, thus to get the least common multiple (LCM), we need to multiply them.
  • Consider the two prime numbers, 'a' and 'b'.

The LCM of 'a' and 'b' is given by

LCM (a, b) = a × b

  • To understand it, consider an example:

Let the two numbers be 7 and 11.

We know that 7 and 11 are prime numbers.

 So, LCM of 7 and 11 = 7 × 11 = 77

LCM of co-prime numbers

  • Since co-prime numbers do not have any common factor other than 1, thus to get the least common multiple, we need to multiply them.
  • Consider a set of two co-prime numbers, (a, b).

Then, LCM (a, b) = a × b

For example:

LCM of co-prime numbers, 4 and 5 is

LCM (4, 5) = 4 × 5 = 20

Illustration Questions

Find the LCM of the two prime numbers, 3 and 13.

A 19

B 29

C 39

D 49

×

Given prime numbers: 3 and 13

The Prime numbers do not have any common factor other than 1, thus to get LCM of 3 and 13, we need to multiply them.

\(\therefore\) The LCM of the prime numbers 3 and 13 = 3 × 13 = 39

Hence, option (C) is correct.

Find the LCM of the two prime numbers, 3 and 13.

A

19

.

B

29

C

39

D

49

Option C is Correct

Words Problems of LCM

  • We know how to find the LCM but we do not know how to use its concept to solve the real life problems.
  • To solve the real world problems, we should know what exactly is being asked in the problem.
  • Problems of LCM may be asking us -
  • About an event or activity which is repeating at the same place after the same time.
  • To get items in order to have enough.

Consider an example:

  • Cody and Jacob start jogging around a park at the same time. Cody finishes one round in \(6\) minutes and Jacob finishes in \(12\) minutes. When will they meet at the starting point again?

In this example, we have to find the time of an event or activity which is repeating again.

Therefore, we should find the LCM of the times in which Cody and Jacob finish one round.

Common factors \(=2\) and \(3\)

Non-common factors \(=2\)

Product of these all factors

\(=2×3×2\)

\(=12\)

Thus, they will meet again at the starting point after \(12\) minutes.

Common factors \(=2\)

Non-common factors \(=3,\;2\) and \(2\)

Product of these all factors

\(=2×3×2×2\)

\(=24\)

Thus, they will meet again at the starting point after \(24\) minutes.

Illustration Questions

Alex exercises after every \(4\) days and Aron exercises after every \(6\) days. Alex and Aron both exercised today. After how many days they will exercise together?

A \(4\)

B \(6\)

C \(8\)

D \(12\)

×

Given:

Alex exercises = after every \(4\) days

Aron exercises = after every \(6\) days

In this problem, we have been asked about an event which is repeating at the same time and same place.

\(\therefore\) We will find the LCM of \(4\) days and \(6\) days.

Prime factorization of \(4\) and \(6\):

image

Common factors \(=2\)

Non-common factors \(=2\) and \(3\)

Product of all these factors

\(=2×2×3\)

\(=12\)

Thus, after \(12\) days they will exercise together again.

Hence, option (D) is correct.

Alex exercises after every \(4\) days and Aron exercises after every \(6\) days. Alex and Aron both exercised today. After how many days they will exercise together?

A

\(4\)

.

B

\(6\)

C

\(8\)

D

\(12\)

Option D is Correct

Least Common Multiple (LCM) by Prime Factorization

  • The LCM of two numbers can be found out with the help of prime factorization also.
  • To get the LCM of two numbers, factorize them with the help of a factor tree to find out their prime factors.
  • We should follow the given steps to find the LCM of the given numbers.

Consider an example:

Find the LCM of \(6\) and \(12\).

Step-1: Find the prime factors of \(6\) and \(12\) with the help of the factor trees as shown.

Prime factors of \(6=2\) and \(3\)

Prime factors of \(12=2,\;2\) and \(3\)

Step-2: Choose the common prime factors between them.

Here, the common prime factors \(=2\) and \(3\)

Step-3: Take those prime factors which are not common between the given numbers.

Here, the non-common prime factor \(=2\)

Step-4: Multiply the common prime factors with all the non-common prime factors, the product of all these factors is the LCM.

So, the product of all these (common and non-common) factors \(=2×2×3\)

\(=12\)

Thus, the LCM of \(6\) and \(12=12\)

Prime factors of \(6=2\) and \(3\)

Prime factors of \(12=2,\;2\) and \(3\)

Step-2 Choose the common prime factors between them.

Here, the common prime factors \(=2\) and \(3\)

Step-3 Take those prime factors which are not common between the given numbers.

Here, the non-common prime factor \(=2\)

Step-4 Multiply the common prime factors with all the non-common prime factors, the product of all these factors is the LCM.

So, the product of all these (common and non-common) factors \(=2×2×3\)

\(=12\)

Thus, the LCM of \(6\) and \(12=12\)

Illustration Questions

What is the LCM of \(9\) and \(12\)?

A \(18\)

B \(36\)

C \(24\)

D \(27\)

×

Given numbers: \(9\) and \(12\)

The prime factors of \(9\) and \(12\) are shown in factor trees.

image

The common prime factor \(=3\)

The non-common prime factors \(=3,\;2\) and \(2\)

LCM = Product of (common prime factors and non-common prime factors)

\(=3×3×2×2\)

\(=36\)

Thus, the LCM of \(9\) and \(12=36\)

Hence, option (B) is correct.

What is the LCM of \(9\) and \(12\)?

A

\(18\)

.

B

\(36\)

C

\(24\)

D

\(27\)

Option B is Correct

Least Common Multiple (LCM)

  • The least common multiple (LCM) is also known as the lowest common multiple or the smallest common multiple.

Definition:  The smallest multiple which is common in two or more than two numbers is called LCM of those numbers.

For example: 

\(6\) is the smallest multiple which is common in \(2\) and \(3\).

So, \(6\) is the LCM of \(2\) and \(3\).

  • We can also define it in terms of divisibility:
  •  The smallest multiple of two or more numbers which is divisible by all the given numbers is their LCM.
  • In the above example, \(6\) is the smallest multiple which is divisible by both \(2\) and \(3\).
  • The LCM of two numbers, a and b is denoted by

\(LCM(a,b)= c\)

Steps to find the LCM: 

  • First, write some multiples of all the given numbers.
  • Select the common multiples from these.
  • Now, choose the lowest common multiple of the numbers.

For example:

We have two numbers, \(2\) and \(3\).

Step 1: The multiples of \(2\;=\;2,4,6,8,10,12,14,16,18,.....\) 

The multiples of \(3\;=\;3,6,9,12,15,18,.......\)

Step 2: The common multiples \(=6,12,18\)

Step 3: The least common multiple \(=6\)

So, the LCM \((2,3)=6\)

Illustration Questions

Find the LCM of \(3\) and \(4\).

A \(12\)

B \(9\)

C \(16\)

D \(24\)

×

The multiples of \(3=3,6,9,12,15,18,21,24,27,.....\)

The multiples of \(4=4,8,12,16,20,24,28,32,.....\)

The common multiples of \(3\) and \(4\) \(=12,24,............\)

The least common multiple \(=12\)

So, the LCM \((3,4)=12\)

Hence, option (A) is correct.

Find the LCM of \(3\) and \(4\).

A

\(12\)

.

B

\(9\)

C

\(16\)

D

\(24\)

Option A is Correct

Relation Between LCM and GCF

  •  The product of LCM and GCF of the given numbers is equal to the product of that numbers.

For example: 

Consider the two numbers, \(6\) and \(9\).

The GCF of  \(6\) and \(9=3\) 

The LCM of  \(6\) and \(9=18\)

The product of LCM and GCF \(=3×18\)

\(=54\)

The product of \(6\) and \(9=6×9\)

\(=54\)

So, \(\text{GCF × LCM = Product of the numbers}\)

  • The GCF of the given numbers is a factor of their LCM.

For example:

  • The GCF of \(4\) and \(12=4\)
  • The LCM of \(4\) and \(12=12\)
  • Here, \(4\) (GCF) is a factor of \(12\) (LCM).

The LCM of the given numbers is a multiple of their GCF.

In the above example:

\(12\) (LCM) is a multiple of \(4\) (GCF).

  • If GCF of two numbers is one of them then the other number is their LCM.

For example:

The GCF of \(2\) and \(6=2\)

So, the LCM of \(2\) and \(6=6\)

Illustration Questions

LCM (a,b) \(=24\) and GCF (a,b)​ \(=2\). Find the value of b, if a \(=6\).

A \(24\)

B \(6\)

C \(12\)

D \(8\)

×

Given:

LCM (a,b) \(=24\) 

GCF (a,b) \(=2\)

\(=6\)

The relation between LCM and GCF is

\(\text{LCM (a,b)}\times \text{GCF (a,b) = a×b} \)

On putting the values, we get:

\(24×2=6×b \)

\(b=\dfrac{24×2}{6}\)

\(b=8\)

Hence, option (D) is correct.

LCM (a,b) \(=24\) and GCF (a,b)​ \(=2\). Find the value of b, if a \(=6\).

A

\(24\)

.

B

\(6\)

C

\(12\)

D

\(8\)

Option D is Correct

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