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LCM of 78 and 104

LCM of 78 and 104 is the smallest number among all common multiples of 78 and 104. The first few multiples of 78 and 104 are (78, 156, 234, 312, . . . ) and (104, 208, 312, 416, 520, 624, . . . ) respectively. There are 3 commonly used methods to find LCM of 78 and 104 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 78 and 104
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 78 and 104?

Answer: LCM of 78 and 104 is 312.

Explanation:

The LCM of two non-zero integers, x(78) and y(104), is the smallest positive integer m(312) that is divisible by both x(78) and y(104) without any remainder.

Methods to Find LCM of 78 and 104

Let's look at the different methods for finding the LCM of 78 and 104.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 78 and 104 by Listing Multiples

To calculate the LCM of 78 and 104 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 78 (78, 156, 234, 312, . . . ) and 104 (104, 208, 312, 416, 520, 624, . . . . )
Step 2: The common multiples from the multiples of 78 and 104 are 312, 624, . . .
Step 3: The smallest common multiple of 78 and 104 is 312.

∴ The least common multiple of 78 and 104 = 312.

LCM of 78 and 104 by Prime Factorization

Prime factorization of 78 and 104 is (2 × 3 × 13) = 2¹ × 3¹ × 13¹ and (2 × 2 × 2 × 13) = 2³ × 13¹ respectively. LCM of 78 and 104 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3¹ × 13¹ = 312.
Hence, the LCM of 78 and 104 by prime factorization is 312.

LCM of 78 and 104 by Division Method

To calculate the LCM of 78 and 104 by the division method, we will divide the numbers(78, 104) by their prime factors (preferably common). The product of these divisors gives the LCM of 78 and 104.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 78 and 104. Write this prime number(2) on the left of the given numbers(78 and 104), separated as per the ladder arrangement.
Step 2: If any of the given numbers (78, 104) 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 78 and 104 is the product of all prime numbers on the left, i.e. LCM(78, 104) by division method = 2 × 2 × 2 × 3 × 13 = 312.

☛ Also Check:

LCM of 78 and 104 Examples

Example 1: The product of two numbers is 8112. If their GCD is 26, what is their LCM?

Solution:

Given: GCD = 26
product of numbers = 8112
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 8112/26
Therefore, the LCM is 312.
The probable combination for the given case is LCM(78, 104) = 312.
Example 2: Verify the relationship between GCF and LCM of 78 and 104.

Solution:

The relation between GCF and LCM of 78 and 104 is given as,
LCM(78, 104) × GCF(78, 104) = Product of 78, 104
Prime factorization of 78 and 104 is given as, 78 = (2 × 3 × 13) = 2¹ × 3¹ × 13¹ and 104 = (2 × 2 × 2 × 13) = 2³ × 13¹
LCM(78, 104) = 312
GCF(78, 104) = 26
LHS = LCM(78, 104) × GCF(78, 104) = 312 × 26 = 8112
RHS = Product of 78, 104 = 78 × 104 = 8112
⇒ LHS = RHS = 8112
Hence, verified.
Example 3: The GCD and LCM of two numbers are 26 and 312 respectively. If one number is 78, find the other number.

Solution:

Let the other number be z.
∵ GCD × LCM = 78 × z
⇒ z = (GCD × LCM)/78
⇒ z = (26 × 312)/78
⇒ z = 104
Therefore, the other number is 104.

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FAQs on LCM of 78 and 104

What is the LCM of 78 and 104?

The LCM of 78 and 104 is 312. To find the least common multiple (LCM) of 78 and 104, we need to find the multiples of 78 and 104 (multiples of 78 = 78, 156, 234, 312; multiples of 104 = 104, 208, 312, 416) and choose the smallest multiple that is exactly divisible by 78 and 104, i.e., 312.

How to Find the LCM of 78 and 104 by Prime Factorization?

To find the LCM of 78 and 104 using prime factorization, we will find the prime factors, (78 = 2 × 3 × 13) and (104 = 2 × 2 × 2 × 13). LCM of 78 and 104 is the product of prime factors raised to their respective highest exponent among the numbers 78 and 104.
⇒ LCM of 78, 104 = 2³ × 3¹ × 13¹ = 312.

What is the Relation Between GCF and LCM of 78, 104?

The following equation can be used to express the relation between GCF and LCM of 78 and 104, i.e. GCF × LCM = 78 × 104.

Which of the following is the LCM of 78 and 104? 27, 36, 312, 24

The value of LCM of 78, 104 is the smallest common multiple of 78 and 104. The number satisfying the given condition is 312.

If the LCM of 104 and 78 is 312, Find its GCF.

LCM(104, 78) × GCF(104, 78) = 104 × 78
Since the LCM of 104 and 78 = 312
⇒ 312 × GCF(104, 78) = 8112
Therefore, the GCF = 8112/312 = 26.

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