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LCM of 144 and 169

LCM of 144 and 169 is the smallest number among all common multiples of 144 and 169. The first few multiples of 144 and 169 are (144, 288, 432, 576, 720, . . . ) and (169, 338, 507, 676, 845, 1014, 1183, . . . ) respectively. There are 3 commonly used methods to find LCM of 144 and 169 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 144 and 169
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 144 and 169?

Answer: LCM of 144 and 169 is 24336.

Explanation:

The LCM of two non-zero integers, x(144) and y(169), is the smallest positive integer m(24336) that is divisible by both x(144) and y(169) without any remainder.

Methods to Find LCM of 144 and 169

The methods to find the LCM of 144 and 169 are explained below.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 144 and 169 by Division Method

To calculate the LCM of 144 and 169 by the division method, we will divide the numbers(144, 169) by their prime factors (preferably common). The product of these divisors gives the LCM of 144 and 169.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 144 and 169. Write this prime number(2) on the left of the given numbers(144 and 169), separated as per the ladder arrangement.
Step 2: If any of the given numbers (144, 169) 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 and 169 is the product of all prime numbers on the left, i.e. LCM(144, 169) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 13 × 13 = 24336.

LCM of 144 and 169 by Listing Multiples

To calculate the LCM of 144 and 169 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, . . . ) and 169 (169, 338, 507, 676, 845, 1014, 1183, . . . . )
Step 2: The common multiples from the multiples of 144 and 169 are 24336, 48672, . . .
Step 3: The smallest common multiple of 144 and 169 is 24336.

∴ The least common multiple of 144 and 169 = 24336.

LCM of 144 and 169 by Prime Factorization

Prime factorization of 144 and 169 is (2 × 2 × 2 × 2 × 3 × 3) = 2⁴ × 3² and (13 × 13) = 13² respectively. LCM of 144 and 169 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3² × 13² = 24336.
Hence, the LCM of 144 and 169 by prime factorization is 24336.

☛ Also Check:

LCM of 144 and 169 Examples

Example 1: The product of two numbers is 24336. If their GCD is 1, what is their LCM?

Solution:

Given: GCD = 1
product of numbers = 24336
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 24336/1
Therefore, the LCM is 24336.
The probable combination for the given case is LCM(144, 169) = 24336.
Example 2: Verify the relationship between GCF and LCM of 144 and 169.

Solution:

The relation between GCF and LCM of 144 and 169 is given as,
LCM(144, 169) × GCF(144, 169) = Product of 144, 169
Prime factorization of 144 and 169 is given as, 144 = (2 × 2 × 2 × 2 × 3 × 3) = 2⁴ × 3² and 169 = (13 × 13) = 13²
LCM(144, 169) = 24336
GCF(144, 169) = 1
LHS = LCM(144, 169) × GCF(144, 169) = 24336 × 1 = 24336
RHS = Product of 144, 169 = 144 × 169 = 24336
⇒ LHS = RHS = 24336
Hence, verified.
Example 3: The GCD and LCM of two numbers are 1 and 24336 respectively. If one number is 144, find the other number.

Solution:

Let the other number be z.
∵ GCD × LCM = 144 × z
⇒ z = (GCD × LCM)/144
⇒ z = (1 × 24336)/144
⇒ z = 169
Therefore, the other number is 169.

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FAQs on LCM of 144 and 169

What is the LCM of 144 and 169?

The LCM of 144 and 169 is 24336. To find the least common multiple (LCM) of 144 and 169, we need to find the multiples of 144 and 169 (multiples of 144 = 144, 288, 432, 576 . . . . 24336; multiples of 169 = 169, 338, 507, 676 . . . . 24336) and choose the smallest multiple that is exactly divisible by 144 and 169, i.e., 24336.

What is the Relation Between GCF and LCM of 144, 169?

The following equation can be used to express the relation between GCF and LCM of 144 and 169, i.e. GCF × LCM = 144 × 169.

What is the Least Perfect Square Divisible by 144 and 169?

The least number divisible by 144 and 169 = LCM(144, 169)
LCM of 144 and 169 = 2 × 2 × 2 × 2 × 3 × 3 × 13 × 13 [No incomplete pair]
⇒ Least perfect square divisible by each 144 and 169 = 24336 [Square root of 24336 = √24336 = ±156]
Therefore, 24336 is the required number.

If the LCM of 169 and 144 is 24336, Find its GCF.

LCM(169, 144) × GCF(169, 144) = 169 × 144
Since the LCM of 169 and 144 = 24336
⇒ 24336 × GCF(169, 144) = 24336
Therefore, the GCF = 24336/24336 = 1.

How to Find the LCM of 144 and 169 by Prime Factorization?

To find the LCM of 144 and 169 using prime factorization, we will find the prime factors, (144 = 2 × 2 × 2 × 2 × 3 × 3) and (169 = 13 × 13). LCM of 144 and 169 is the product of prime factors raised to their respective highest exponent among the numbers 144 and 169.
⇒ LCM of 144, 169 = 2⁴ × 3² × 13² = 24336.

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