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LCM of 40, 42, and 45

LCM of 40, 42, and 45 is the smallest number among all common multiples of 40, 42, and 45. The first few multiples of 40, 42, and 45 are (40, 80, 120, 160, 200 . . .), (42, 84, 126, 168, 210 . . .), and (45, 90, 135, 180, 225 . . .) respectively. There are 3 commonly used methods to find LCM of 40, 42, 45 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 40, 42, and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 40, 42, and 45?

Answer: LCM of 40, 42, and 45 is 2520.

Explanation:

The LCM of three non-zero integers, a(40), b(42), and c(45), is the smallest positive integer m(2520) that is divisible by a(40), b(42), and c(45) without any remainder.

Methods to Find LCM of 40, 42, and 45

Let's look at the different methods for finding the LCM of 40, 42, and 45.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 40, 42, and 45 by Division Method

To calculate the LCM of 40, 42, and 45 by the division method, we will divide the numbers(40, 42, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 40, 42, and 45.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 40, 42, and 45. Write this prime number(2) on the left of the given numbers(40, 42, and 45), separated as per the ladder arrangement.
Step 2: If any of the given numbers (40, 42, 45) 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 40, 42, and 45 is the product of all prime numbers on the left, i.e. LCM(40, 42, 45) by division method = 2 × 2 × 2 × 3 × 3 × 5 × 7 = 2520.

LCM of 40, 42, and 45 by Listing Multiples

To calculate the LCM of 40, 42, 45 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 40 (40, 80, 120, 160, 200 . . .), 42 (42, 84, 126, 168, 210 . . .), and 45 (45, 90, 135, 180, 225 . . .).
Step 2: The common multiples from the multiples of 40, 42, and 45 are 2520, 5040, . . .
Step 3: The smallest common multiple of 40, 42, and 45 is 2520.

∴ The least common multiple of 40, 42, and 45 = 2520.

LCM of 40, 42, and 45 by Prime Factorization

Prime factorization of 40, 42, and 45 is (2 × 2 × 2 × 5) = 2³ × 5¹, (2 × 3 × 7) = 2¹ × 3¹ × 7¹, and (3 × 3 × 5) = 3² × 5¹ respectively. LCM of 40, 42, and 45 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ × 7¹ = 2520.
Hence, the LCM of 40, 42, and 45 by prime factorization is 2520.

☛ Also Check:

LCM of 40, 42, and 45 Examples

Example 1: Verify the relationship between the GCD and LCM of 40, 42, and 45.

Solution:

The relation between GCD and LCM of 40, 42, and 45 is given as,
LCM(40, 42, 45) = [(40 × 42 × 45) × GCD(40, 42, 45)]/[GCD(40, 42) × GCD(42, 45) × GCD(40, 45)]
⇒ Prime factorization of 40, 42 and 45:
- 40 = 2³ × 5¹
- 42 = 2¹ × 3¹ × 7¹
- 45 = 3² × 5¹
∴ GCD of (40, 42), (42, 45), (40, 45) and (40, 42, 45) = 2, 3, 5 and 1 respectively.
Now, LHS = LCM(40, 42, 45) = 2520.
And, RHS = [(40 × 42 × 45) × GCD(40, 42, 45)]/[GCD(40, 42) × GCD(42, 45) × GCD(40, 45)] = [(75600) × 1]/[2 × 3 × 5] = 2520
LHS = RHS = 2520.
Hence verified.
Example 2: Find the smallest number that is divisible by 40, 42, 45 exactly.

Solution:

The value of LCM(40, 42, 45) will be the smallest number that is exactly divisible by 40, 42, and 45.
⇒ Multiples of 40, 42, and 45:
- Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, . . . ., 2440, 2480, 2520, . . . .
- Multiples of 42 = 42, 84, 126, 168, 210, 252, 294, 336, 378, 420, . . . ., 2436, 2478, 2520, . . . .
- Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, . . . ., 2430, 2475, 2520, . . . .
Therefore, the LCM of 40, 42, and 45 is 2520.
Example 3: Calculate the LCM of 40, 42, and 45 using the GCD of the given numbers.

Solution:

Prime factorization of 40, 42, 45:
- 40 = 2³ × 5¹
- 42 = 2¹ × 3¹ × 7¹
- 45 = 3² × 5¹
Therefore, GCD(40, 42) = 2, GCD(42, 45) = 3, GCD(40, 45) = 5, GCD(40, 42, 45) = 1
We know,
LCM(40, 42, 45) = [(40 × 42 × 45) × GCD(40, 42, 45)]/[GCD(40, 42) × GCD(42, 45) × GCD(40, 45)]
LCM(40, 42, 45) = (75600 × 1)/(2 × 3 × 5) = 2520
⇒LCM(40, 42, 45) = 2520

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FAQs on LCM of 40, 42, and 45

What is the LCM of 40, 42, and 45?

The LCM of 40, 42, and 45 is 2520. To find the LCM (least common multiple) of 40, 42, and 45, we need to find the multiples of 40, 42, and 45 (multiples of 40 = 40, 80, 120, 160 . . . . 2520 . . . . ; multiples of 42 = 42, 84, 126, 168 . . . . 2520 . . . . ; multiples of 45 = 45, 90, 135, 180 . . . . 2520 . . . . ) and choose the smallest multiple that is exactly divisible by 40, 42, and 45, i.e., 2520.

What is the Least Perfect Square Divisible by 40, 42, and 45?

The least number divisible by 40, 42, and 45 = LCM(40, 42, 45)
LCM of 40, 42, and 45 = 2 × 2 × 2 × 3 × 3 × 5 × 7 [Incomplete pair(s): 2, 5, 7]
⇒ Least perfect square divisible by each 40, 42, and 45 = LCM(40, 42, 45) × 2 × 5 × 7 = 176400 [Square root of 176400 = √176400 = ±420]
Therefore, 176400 is the required number.

What is the Relation Between GCF and LCM of 40, 42, 45?

The following equation can be used to express the relation between GCF and LCM of 40, 42, 45, i.e. LCM(40, 42, 45) = [(40 × 42 × 45) × GCF(40, 42, 45)]/[GCF(40, 42) × GCF(42, 45) × GCF(40, 45)].

Which of the following is the LCM of 40, 42, and 45? 2520, 96, 20, 11

The value of LCM of 40, 42, 45 is the smallest common multiple of 40, 42, and 45. The number satisfying the given condition is 2520.

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