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LCM of 40 and 56

LCM of 40 and 56 is the smallest number among all common multiples of 40 and 56. The first few multiples of 40 and 56 are (40, 80, 120, 160, 200, 240, . . . ) and (56, 112, 168, 224, 280, 336, . . . ) respectively. There are 3 commonly used methods to find LCM of 40 and 56 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 40 and 56
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 40 and 56?

Answer: LCM of 40 and 56 is 280.

Explanation:

The LCM of two non-zero integers, x(40) and y(56), is the smallest positive integer m(280) that is divisible by both x(40) and y(56) without any remainder.

Methods to Find LCM of 40 and 56

The methods to find the LCM of 40 and 56 are explained below.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 40 and 56 by Prime Factorization

Prime factorization of 40 and 56 is (2 × 2 × 2 × 5) = 2³ × 5¹ and (2 × 2 × 2 × 7) = 2³ × 7¹ respectively. LCM of 40 and 56 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 5¹ × 7¹ = 280.
Hence, the LCM of 40 and 56 by prime factorization is 280.

LCM of 40 and 56 by Division Method

To calculate the LCM of 40 and 56 by the division method, we will divide the numbers(40, 56) by their prime factors (preferably common). The product of these divisors gives the LCM of 40 and 56.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 40 and 56. Write this prime number(2) on the left of the given numbers(40 and 56), separated as per the ladder arrangement.
Step 2: If any of the given numbers (40, 56) 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 and 56 is the product of all prime numbers on the left, i.e. LCM(40, 56) by division method = 2 × 2 × 2 × 5 × 7 = 280.

LCM of 40 and 56 by Listing Multiples

To calculate the LCM of 40 and 56 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, 240, . . . ) and 56 (56, 112, 168, 224, 280, 336, . . . . )
Step 2: The common multiples from the multiples of 40 and 56 are 280, 560, . . .
Step 3: The smallest common multiple of 40 and 56 is 280.

∴ The least common multiple of 40 and 56 = 280.

☛ Also Check:

LCM of 40 and 56 Examples

Example 1: Find the smallest number that is divisible by 40 and 56 exactly.

Solution:

The smallest number that is divisible by 40 and 56 exactly is their LCM.
⇒ Multiples of 40 and 56:
- Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, . . . .
- Multiples of 56 = 56, 112, 168, 224, 280, . . . .
Therefore, the LCM of 40 and 56 is 280.
Example 2: Verify the relationship between GCF and LCM of 40 and 56.

Solution:

The relation between GCF and LCM of 40 and 56 is given as,
LCM(40, 56) × GCF(40, 56) = Product of 40, 56
Prime factorization of 40 and 56 is given as, 40 = (2 × 2 × 2 × 5) = 2³ × 5¹ and 56 = (2 × 2 × 2 × 7) = 2³ × 7¹
LCM(40, 56) = 280
GCF(40, 56) = 8
LHS = LCM(40, 56) × GCF(40, 56) = 280 × 8 = 2240
RHS = Product of 40, 56 = 40 × 56 = 2240
⇒ LHS = RHS = 2240
Hence, verified.
Example 3: The GCD and LCM of two numbers are 8 and 280 respectively. If one number is 40, find the other number.

Solution:

Let the other number be m.
∵ GCD × LCM = 40 × m
⇒ m = (GCD × LCM)/40
⇒ m = (8 × 280)/40
⇒ m = 56
Therefore, the other number is 56.

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FAQs on LCM of 40 and 56

What is the LCM of 40 and 56?

The LCM of 40 and 56 is 280. To find the LCM (least common multiple) of 40 and 56, we need to find the multiples of 40 and 56 (multiples of 40 = 40, 80, 120, 160 . . . . 280; multiples of 56 = 56, 112, 168, 224 . . . . 280) and choose the smallest multiple that is exactly divisible by 40 and 56, i.e., 280.

If the LCM of 56 and 40 is 280, Find its GCF.

LCM(56, 40) × GCF(56, 40) = 56 × 40
Since the LCM of 56 and 40 = 280
⇒ 280 × GCF(56, 40) = 2240
Therefore, the GCF = 2240/280 = 8.

How to Find the LCM of 40 and 56 by Prime Factorization?

To find the LCM of 40 and 56 using prime factorization, we will find the prime factors, (40 = 2 × 2 × 2 × 5) and (56 = 2 × 2 × 2 × 7). LCM of 40 and 56 is the product of prime factors raised to their respective highest exponent among the numbers 40 and 56.
⇒ LCM of 40, 56 = 2³ × 5¹ × 7¹ = 280.

Which of the following is the LCM of 40 and 56? 18, 42, 280, 32

The value of LCM of 40, 56 is the smallest common multiple of 40 and 56. The number satisfying the given condition is 280.

What is the Least Perfect Square Divisible by 40 and 56?

The least number divisible by 40 and 56 = LCM(40, 56)
LCM of 40 and 56 = 2 × 2 × 2 × 5 × 7 [Incomplete pair(s): 2, 5, 7]
⇒ Least perfect square divisible by each 40 and 56 = LCM(40, 56) × 2 × 5 × 7 = 19600 [Square root of 19600 = √19600 = ±140]
Therefore, 19600 is the required number.

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