Find the LCM of two given numbers 144 and 192

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Find the lcm and hcf of 180, 144 and 192 by applying prime factorization method Maths Q&A

Last updated - March 20, 2022

Answer: HCF = 48 and LCM = 576

Step by step solution:

Contents:

Given numbers = 144 and 192

To find HCF and LCM by prime factorization method, first we will find prime factors of given numbers.

Prime Factorization of 144:
144 = 2 × 72
= 2 × 2 × 36
= 2 × 2 × 2 × 18
= 2 × 2 × 2 × 2 × 9
= 2 × 2 × 2 × 2 × 3 × 3

Prime Factorization of 192:

HCF of 144 and 192 by prime factorization method:

Common factors in above prime factors of given numbers are underlined.

Common prime factors = 2, 2, 2, 2, 3

Now we have to multiply these common prime factors to obtain the HCF of given numbers.

HCF = 2 × 2 × 2 × 2 × 3
= 48

∴ HCF(144, 192) = 48

LCM of 144 and 192 by prime factorization method:

Now to find the LCM we will note down how many times the each factor has occurred in above prime factors of given numbers.

In above table we have also noted the maximum occurrence of the each factor in the prime factors of the given numbers.

Now to obtain the LCM of given numbers we will multiply each factor maximum number of times it occurred in above table.

LCM = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
= 576

∴ LCM(144, 192) = 576

Division Method:

HCF of 144 and 192 by division method:

In the above division, the last divisor is 48.

Hence the HCF(GCF) of 144 and 192 = 48

∴ HCF(144, 192) = 48

LCM of 144 and 192 by division method

∴ LCM(144, 192) = 576

HCF of 144 and 192 by Listing Factors Method:

To find HCF by listing factors method we will list down all the factors of the given numbers.

Factors of 144:

123468912161824364872144

Factors of 192:

12346812162432486496192

The greatest common factor in the above lists will be the HCF of the given numbers.

48 is the greatest factor which is common in the above lists.

∴ HCF(144, 192) = 48

LCM of 144 and 192 by Listing Multiples Method:

To find LCM by listing multiples method we will list down the multiples of given numbers.

Multiples of 144:

1442884325767208641008115212961440

Multiples of 192:

19238457676896011521344153617281920

The lowest common multiple in the above lists will be the LCM of the given numbers.

576 is the lowest multiple which is common in the above lists.

∴ LCM(144, 192) = 576

If the HCF of two numbers is 48 and their product is 27648, what is their LCM?

If the LCM of two numbers is 576 and their product is 27648, what is their HCF?

HCF and LCM of two numbers is 48 and 576 respectively. If one number is 192, what is the other number?

Let other number be x.
Given:
HCF(192, x) = 48
LCM(192, x) = 576
∵ HCF × LCM = product of numbers
∴ 48 × 576 = 192 × x
∴ 27648 = 192x
∴ 27648⁄192 = x
∴ x = 144
Hence, the other number is 144.

LCM of 144, 180, and 192 is the smallest number among all common multiples of 144, 180, and 192. The first few multiples of 144, 180, and 192 are (144, 288, 432, 576, 720 . . .), (180, 360, 540, 720, 900 . . .), and (192, 384, 576, 768, 960 . . .) respectively. There are 3 commonly used methods to find LCM of 144, 180, 192 - by division method, by prime factorization, and by listing multiples.

What is the LCM of 144, 180, and 192?

Answer: LCM of 144, 180, and 192 is 2880.

Explanation:

The LCM of three non-zero integers, a(144), b(180), and c(192), is the smallest positive integer m(2880) that is divisible by a(144), b(180), and c(192) without any remainder.

Methods to Find LCM of 144, 180, and 192

The methods to find the LCM of 144, 180, and 192 are explained below.

• By Prime Factorization Method
• By Division Method
• By Listing Multiples

LCM of 144, 180, and 192 by Prime Factorization

Prime factorization of 144, 180, and 192 is (2 × 2 × 2 × 2 × 3 × 3) = 24 × 32, (2 × 2 × 3 × 3 × 5) = 22 × 32 × 51, and (2 × 2 × 2 × 2 × 2 × 2 × 3) = 26 × 31 respectively. LCM of 144, 180, and 192 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 26 × 32 × 51 = 2880.
Hence, the LCM of 144, 180, and 192 by prime factorization is 2880.

LCM of 144, 180, and 192 by Division Method

To calculate the LCM of 144, 180, and 192 by the division method, we will divide the numbers(144, 180, 192) by their prime factors (preferably common). The product of these divisors gives the LCM of 144, 180, and 192.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 144, 180, and 192. Write this prime number(2) on the left of the given numbers(144, 180, and 192), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (144, 180, 192) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
• Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 144, 180, and 192 is the product of all prime numbers on the left, i.e. LCM(144, 180, 192) by division method = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 2880.

LCM of 144, 180, and 192 by Listing Multiples

To calculate the LCM of 144, 180, 192 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 144 (144, 288, 432, 576, 720 . . .), 180 (180, 360, 540, 720, 900 . . .), and 192 (192, 384, 576, 768, 960 . . .).
• Step 2: The common multiples from the multiples of 144, 180, and 192 are 2880, 5760, . . .
• Step 3: The smallest common multiple of 144, 180, and 192 is 2880.

∴ The least common multiple of 144, 180, and 192 = 2880.

☛ Also Check:

LCM of 144, 180, and 192 Examples

1. Example 1: Find the smallest number that is divisible by 144, 180, 192 exactly.

   Solution:

   The value of LCM(144, 180, 192) will be the smallest number that is exactly divisible by 144, 180, and 192.
   ⇒ Multiples of 144, 180, and 192:

   • Multiples of 144 = 144, 288, 432, 576, 720, 864, 1008, 1152, 1296, 1440, . . . ., 2448, 2592, 2736, 2880, . . . .
   • Multiples of 180 = 180, 360, 540, 720, 900, 1080, 1260, 1440, 1620, 1800, . . . ., 2340, 2520, 2700, 2880, . . . .
   • Multiples of 192 = 192, 384, 576, 768, 960, 1152, 1344, 1536, 1728, 1920, . . . ., 2304, 2496, 2688, 2880, . . . .

   Therefore, the LCM of 144, 180, and 192 is 2880.

2. Example 2: Verify the relationship between the GCD and LCM of 144, 180, and 192.

   Solution:

   The relation between GCD and LCM of 144, 180, and 192 is given as, LCM(144, 180, 192) = [(144 × 180 × 192) × GCD(144, 180, 192)]/[GCD(144, 180) × GCD(180, 192) × GCD(144, 192)]

   ⇒ Prime factorization of 144, 180 and 192:

• 144 = 24 × 32
• 180 = 22 × 32 × 51
• 192 = 26 × 31

∴ GCD of (144, 180), (180, 192), (144, 192) and (144, 180, 192) = 36, 12, 48 and 12 respectively. Now, LHS = LCM(144, 180, 192) = 2880. And, RHS = [(144 × 180 × 192) × GCD(144, 180, 192)]/[GCD(144, 180) × GCD(180, 192) × GCD(144, 192)] = [(4976640) × 12]/[36 × 12 × 48] = 2880 LHS = RHS = 2880.

Hence verified.

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The LCM of 144, 180, and 192 is 2880. To find the LCM (least common multiple) of 144, 180, and 192, we need to find the multiples of 144, 180, and 192 (multiples of 144 = 144, 288, 432, 576 . . . . 2880 . . . . ; multiples of 180 = 180, 360, 540, 720 . . . . 2880 . . . . ; multiples of 192 = 192, 384, 576, 768 . . . . 2880 . . . . ) and choose the smallest multiple that is exactly divisible by 144, 180, and 192, i.e., 2880.

What are the Methods to Find LCM of 144, 180, 192?

The commonly used methods to find the LCM of 144, 180, 192 are:

• Division Method
• Prime Factorization Method
• Listing Multiples

What is the Relation Between GCF and LCM of 144, 180, 192?

The following equation can be used to express the relation between GCF and LCM of 144, 180, 192, i.e. LCM(144, 180, 192) = [(144 × 180 × 192) × GCF(144, 180, 192)]/[GCF(144, 180) × GCF(180, 192) × GCF(144, 192)].

How to Find the LCM of 144, 180, and 192 by Prime Factorization?

To find the LCM of 144, 180, and 192 using prime factorization, we will find the prime factors, (144 = 24 × 32), (180 = 22 × 32 × 51), and (192 = 26 × 31). LCM of 144, 180, and 192 is the product of prime factors raised to their respective highest exponent among the numbers 144, 180, and 192.
⇒ LCM of 144, 180, 192 = 26 × 32 × 51 = 2880.