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LCM of 70, 105, and 175

LCM of 70, 105, and 175 is the smallest number among all common multiples of 70, 105, and 175. The first few multiples of 70, 105, and 175 are (70, 140, 210, 280, 350 . . .), (105, 210, 315, 420, 525 . . .), and (175, 350, 525, 700, 875 . . .) respectively. There are 3 commonly used methods to find LCM of 70, 105, 175 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 70, 105, and 175
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 70, 105, and 175?

Answer: LCM of 70, 105, and 175 is 1050.

Explanation:

The LCM of three non-zero integers, a(70), b(105), and c(175), is the smallest positive integer m(1050) that is divisible by a(70), b(105), and c(175) without any remainder.

Methods to Find LCM of 70, 105, and 175

Let's look at the different methods for finding the LCM of 70, 105, and 175.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 70, 105, and 175 by Division Method

To calculate the LCM of 70, 105, and 175 by the division method, we will divide the numbers(70, 105, 175) by their prime factors (preferably common). The product of these divisors gives the LCM of 70, 105, and 175.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 70, 105, and 175. Write this prime number(2) on the left of the given numbers(70, 105, and 175), separated as per the ladder arrangement.
Step 2: If any of the given numbers (70, 105, 175) 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 70, 105, and 175 is the product of all prime numbers on the left, i.e. LCM(70, 105, 175) by division method = 2 × 3 × 5 × 5 × 7 = 1050.

LCM of 70, 105, and 175 by Prime Factorization

Prime factorization of 70, 105, and 175 is (2 × 5 × 7) = 2¹ × 5¹ × 7¹, (3 × 5 × 7) = 3¹ × 5¹ × 7¹, and (5 × 5 × 7) = 5² × 7¹ respectively. LCM of 70, 105, and 175 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3¹ × 5² × 7¹ = 1050.
Hence, the LCM of 70, 105, and 175 by prime factorization is 1050.

LCM of 70, 105, and 175 by Listing Multiples

To calculate the LCM of 70, 105, 175 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 70 (70, 140, 210, 280, 350 . . .), 105 (105, 210, 315, 420, 525 . . .), and 175 (175, 350, 525, 700, 875 . . .).
Step 2: The common multiples from the multiples of 70, 105, and 175 are 1050, 2100, . . .
Step 3: The smallest common multiple of 70, 105, and 175 is 1050.

∴ The least common multiple of 70, 105, and 175 = 1050.

☛ Also Check:

LCM of 70, 105, and 175 Examples

Example 1: Verify the relationship between the GCD and LCM of 70, 105, and 175.

Solution:

The relation between GCD and LCM of 70, 105, and 175 is given as,
LCM(70, 105, 175) = [(70 × 105 × 175) × GCD(70, 105, 175)]/[GCD(70, 105) × GCD(105, 175) × GCD(70, 175)]
⇒ Prime factorization of 70, 105 and 175:
- 70 = 2¹ × 5¹ × 7¹
- 105 = 3¹ × 5¹ × 7¹
- 175 = 5² × 7¹
∴ GCD of (70, 105), (105, 175), (70, 175) and (70, 105, 175) = 35, 35, 35 and 35 respectively.
Now, LHS = LCM(70, 105, 175) = 1050.
And, RHS = [(70 × 105 × 175) × GCD(70, 105, 175)]/[GCD(70, 105) × GCD(105, 175) × GCD(70, 175)] = [(1286250) × 35]/[35 × 35 × 35] = 1050
LHS = RHS = 1050.
Hence verified.
Example 2: Find the smallest number that is divisible by 70, 105, 175 exactly.

Solution:

The smallest number that is divisible by 70, 105, and 175 exactly is their LCM.
⇒ Multiples of 70, 105, and 175:
- Multiples of 70 = 70, 140, 210, 280, 350, 420, 490, 560, 630, 700, 770, 840, 910, 980, 1050, . . . .
- Multiples of 105 = 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050, . . . .
- Multiples of 175 = 175, 350, 525, 700, 875, 1050, . . . .
Therefore, the LCM of 70, 105, and 175 is 1050.
Example 3: Calculate the LCM of 70, 105, and 175 using the GCD of the given numbers.

Solution:

Prime factorization of 70, 105, 175:
- 70 = 2¹ × 5¹ × 7¹
- 105 = 3¹ × 5¹ × 7¹
- 175 = 5² × 7¹
Therefore, GCD(70, 105) = 35, GCD(105, 175) = 35, GCD(70, 175) = 35, GCD(70, 105, 175) = 35
We know,
LCM(70, 105, 175) = [(70 × 105 × 175) × GCD(70, 105, 175)]/[GCD(70, 105) × GCD(105, 175) × GCD(70, 175)]
LCM(70, 105, 175) = (1286250 × 35)/(35 × 35 × 35) = 1050
⇒LCM(70, 105, 175) = 1050

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FAQs on LCM of 70, 105, and 175

What is the LCM of 70, 105, and 175?

The LCM of 70, 105, and 175 is 1050. To find the least common multiple (LCM) of 70, 105, and 175, we need to find the multiples of 70, 105, and 175 (multiples of 70 = 70, 140, 210, 280 . . . . 1050 . . . . ; multiples of 105 = 105, 210, 315, 420 . . . . 1050 . . . . ; multiples of 175 = 175, 350, 525, 700, 1050 . . . .) and choose the smallest multiple that is exactly divisible by 70, 105, and 175, i.e., 1050.

What is the Least Perfect Square Divisible by 70, 105, and 175?

The least number divisible by 70, 105, and 175 = LCM(70, 105, 175)
LCM of 70, 105, and 175 = 2 × 3 × 5 × 5 × 7 [Incomplete pair(s): 2, 3, 7]
⇒ Least perfect square divisible by each 70, 105, and 175 = LCM(70, 105, 175) × 2 × 3 × 7 = 44100 [Square root of 44100 = √44100 = ±210]
Therefore, 44100 is the required number.

How to Find the LCM of 70, 105, and 175 by Prime Factorization?

To find the LCM of 70, 105, and 175 using prime factorization, we will find the prime factors, (70 = 2¹ × 5¹ × 7¹), (105 = 3¹ × 5¹ × 7¹), and (175 = 5² × 7¹). LCM of 70, 105, and 175 is the product of prime factors raised to their respective highest exponent among the numbers 70, 105, and 175.
⇒ LCM of 70, 105, 175 = 2¹ × 3¹ × 5² × 7¹ = 1050.

What is the Relation Between GCF and LCM of 70, 105, 175?

The following equation can be used to express the relation between GCF and LCM of 70, 105, 175, i.e. LCM(70, 105, 175) = [(70 × 105 × 175) × GCF(70, 105, 175)]/[GCF(70, 105) × GCF(105, 175) × GCF(70, 175)].

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