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LCM of 35, 12, and 70

LCM of 35, 12, and 70 is the smallest number among all common multiples of 35, 12, and 70. The first few multiples of 35, 12, and 70 are (35, 70, 105, 140, 175 . . .), (12, 24, 36, 48, 60 . . .), and (70, 140, 210, 280, 350 . . .) respectively. There are 3 commonly used methods to find LCM of 35, 12, 70 - by division method, by prime factorization, and by listing multiples.

1.	LCM of 35, 12, and 70
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 35, 12, and 70?

Answer: LCM of 35, 12, and 70 is 420.

Explanation:

The LCM of three non-zero integers, a(35), b(12), and c(70), is the smallest positive integer m(420) that is divisible by a(35), b(12), and c(70) without any remainder.

Methods to Find LCM of 35, 12, and 70

Let's look at the different methods for finding the LCM of 35, 12, and 70.

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 35, 12, and 70 by Listing Multiples

To calculate the LCM of 35, 12, 70 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 35 (35, 70, 105, 140, 175 . . .), 12 (12, 24, 36, 48, 60 . . .), and 70 (70, 140, 210, 280, 350 . . .).
Step 2: The common multiples from the multiples of 35, 12, and 70 are 420, 840, . . .
Step 3: The smallest common multiple of 35, 12, and 70 is 420.

∴ The least common multiple of 35, 12, and 70 = 420.

LCM of 35, 12, and 70 by Division Method

To calculate the LCM of 35, 12, and 70 by the division method, we will divide the numbers(35, 12, 70) by their prime factors (preferably common). The product of these divisors gives the LCM of 35, 12, and 70.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 35, 12, and 70. Write this prime number(2) on the left of the given numbers(35, 12, and 70), separated as per the ladder arrangement.
Step 2: If any of the given numbers (35, 12, 70) 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 35, 12, and 70 is the product of all prime numbers on the left, i.e. LCM(35, 12, 70) by division method = 2 × 2 × 3 × 5 × 7 = 420.

LCM of 35, 12, and 70 by Prime Factorization

Prime factorization of 35, 12, and 70 is (5 × 7) = 5¹ × 7¹, (2 × 2 × 3) = 2² × 3¹, and (2 × 5 × 7) = 2¹ × 5¹ × 7¹ respectively. LCM of 35, 12, and 70 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3¹ × 5¹ × 7¹ = 420.
Hence, the LCM of 35, 12, and 70 by prime factorization is 420.

☛ Also Check:

LCM of 35, 12, and 70 Examples

Example 1: Calculate the LCM of 35, 12, and 70 using the GCD of the given numbers.

Solution:

Prime factorization of 35, 12, 70:
- 35 = 5¹ × 7¹
- 12 = 2² × 3¹
- 70 = 2¹ × 5¹ × 7¹
Therefore, GCD(35, 12) = 1, GCD(12, 70) = 2, GCD(35, 70) = 35, GCD(35, 12, 70) = 1
We know,
LCM(35, 12, 70) = [(35 × 12 × 70) × GCD(35, 12, 70)]/[GCD(35, 12) × GCD(12, 70) × GCD(35, 70)]
LCM(35, 12, 70) = (29400 × 1)/(1 × 2 × 35) = 420
⇒LCM(35, 12, 70) = 420
Example 2: Find the smallest number that is divisible by 35, 12, 70 exactly.

Solution:

The value of LCM(35, 12, 70) will be the smallest number that is exactly divisible by 35, 12, and 70.
⇒ Multiples of 35, 12, and 70:
- Multiples of 35 = 35, 70, 105, 140, 175, 210, 245, 280, 315, 350, . . . ., 315, 350, 385, 420, . . . .
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . ., 372, 384, 396, 408, 420, . . . .
- Multiples of 70 = 70, 140, 210, 280, 350, 420, 490, 560, 630, 700, . . . ., 140, 210, 280, 350, 420, . . . .
Therefore, the LCM of 35, 12, and 70 is 420.
Example 3: Verify the relationship between the GCD and LCM of 35, 12, and 70.

Solution:

The relation between GCD and LCM of 35, 12, and 70 is given as,
LCM(35, 12, 70) = [(35 × 12 × 70) × GCD(35, 12, 70)]/[GCD(35, 12) × GCD(12, 70) × GCD(35, 70)]
⇒ Prime factorization of 35, 12 and 70:
- 35 = 5¹ × 7¹
- 12 = 2² × 3¹
- 70 = 2¹ × 5¹ × 7¹
∴ GCD of (35, 12), (12, 70), (35, 70) and (35, 12, 70) = 1, 2, 35 and 1 respectively.
Now, LHS = LCM(35, 12, 70) = 420.
And, RHS = [(35 × 12 × 70) × GCD(35, 12, 70)]/[GCD(35, 12) × GCD(12, 70) × GCD(35, 70)] = [(29400) × 1]/[1 × 2 × 35] = 420
LHS = RHS = 420.
Hence verified.

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FAQs on LCM of 35, 12, and 70

What is the LCM of 35, 12, and 70?

The LCM of 35, 12, and 70 is 420. To find the least common multiple (LCM) of 35, 12, and 70, we need to find the multiples of 35, 12, and 70 (multiples of 35 = 35, 70, 105, 140 . . . . 420 . . . . ; multiples of 12 = 12, 24, 36, 48 . . . . 420 . . . . ; multiples of 70 = 70, 140, 210, 280, 420 . . . .) and choose the smallest multiple that is exactly divisible by 35, 12, and 70, i.e., 420.

How to Find the LCM of 35, 12, and 70 by Prime Factorization?

To find the LCM of 35, 12, and 70 using prime factorization, we will find the prime factors, (35 = 5¹ × 7¹), (12 = 2² × 3¹), and (70 = 2¹ × 5¹ × 7¹). LCM of 35, 12, and 70 is the product of prime factors raised to their respective highest exponent among the numbers 35, 12, and 70.
⇒ LCM of 35, 12, 70 = 2² × 3¹ × 5¹ × 7¹ = 420.

Which of the following is the LCM of 35, 12, and 70? 15, 40, 50, 420

The value of LCM of 35, 12, 70 is the smallest common multiple of 35, 12, and 70. The number satisfying the given condition is 420.

What is the Relation Between GCF and LCM of 35, 12, 70?

The following equation can be used to express the relation between GCF and LCM of 35, 12, 70, i.e. LCM(35, 12, 70) = [(35 × 12 × 70) × GCF(35, 12, 70)]/[GCF(35, 12) × GCF(12, 70) × GCF(35, 70)].

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