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LCM of 63, 70, and 77

LCM of 63, 70, and 77 is the smallest number among all common multiples of 63, 70, and 77. The first few multiples of 63, 70, and 77 are (63, 126, 189, 252, 315 . . .), (70, 140, 210, 280, 350 . . .), and (77, 154, 231, 308, 385 . . .) respectively. There are 3 commonly used methods to find LCM of 63, 70, 77 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 63, 70, and 77
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 63, 70, and 77?

Answer: LCM of 63, 70, and 77 is 6930.

Explanation:

The LCM of three non-zero integers, a(63), b(70), and c(77), is the smallest positive integer m(6930) that is divisible by a(63), b(70), and c(77) without any remainder.

Methods to Find LCM of 63, 70, and 77

The methods to find the LCM of 63, 70, and 77 are explained below.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 63, 70, and 77 by Prime Factorization

Prime factorization of 63, 70, and 77 is (3 × 3 × 7) = 3² × 7¹, (2 × 5 × 7) = 2¹ × 5¹ × 7¹, and (7 × 11) = 7¹ × 11¹ respectively. LCM of 63, 70, and 77 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3² × 5¹ × 7¹ × 11¹ = 6930.
Hence, the LCM of 63, 70, and 77 by prime factorization is 6930.

LCM of 63, 70, and 77 by Listing Multiples

To calculate the LCM of 63, 70, 77 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 63 (63, 126, 189, 252, 315 . . .), 70 (70, 140, 210, 280, 350 . . .), and 77 (77, 154, 231, 308, 385 . . .).
Step 2: The common multiples from the multiples of 63, 70, and 77 are 6930, 13860, . . .
Step 3: The smallest common multiple of 63, 70, and 77 is 6930.

∴ The least common multiple of 63, 70, and 77 = 6930.

LCM of 63, 70, and 77 by Division Method

To calculate the LCM of 63, 70, and 77 by the division method, we will divide the numbers(63, 70, 77) by their prime factors (preferably common). The product of these divisors gives the LCM of 63, 70, and 77.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 63, 70, and 77. Write this prime number(2) on the left of the given numbers(63, 70, and 77), separated as per the ladder arrangement.
Step 2: If any of the given numbers (63, 70, 77) 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 63, 70, and 77 is the product of all prime numbers on the left, i.e. LCM(63, 70, 77) by division method = 2 × 3 × 3 × 5 × 7 × 11 = 6930.

☛ Also Check:

LCM of 63, 70, and 77 Examples

Example 1: Verify the relationship between the GCD and LCM of 63, 70, and 77.

Solution:

The relation between GCD and LCM of 63, 70, and 77 is given as,
LCM(63, 70, 77) = [(63 × 70 × 77) × GCD(63, 70, 77)]/[GCD(63, 70) × GCD(70, 77) × GCD(63, 77)]
⇒ Prime factorization of 63, 70 and 77:
- 63 = 3² × 7¹
- 70 = 2¹ × 5¹ × 7¹
- 77 = 7¹ × 11¹
∴ GCD of (63, 70), (70, 77), (63, 77) and (63, 70, 77) = 7, 7, 7 and 7 respectively.
Now, LHS = LCM(63, 70, 77) = 6930.
And, RHS = [(63 × 70 × 77) × GCD(63, 70, 77)]/[GCD(63, 70) × GCD(70, 77) × GCD(63, 77)] = [(339570) × 7]/[7 × 7 × 7] = 6930
LHS = RHS = 6930.
Hence verified.
Example 2: Find the smallest number that is divisible by 63, 70, 77 exactly.

Solution:

The value of LCM(63, 70, 77) will be the smallest number that is exactly divisible by 63, 70, and 77.
⇒ Multiples of 63, 70, and 77:
- Multiples of 63 = 63, 126, 189, 252, 315, 378, 441, 504, 567, 630, . . . ., 6741, 6804, 6867, 6930, . . . .
- Multiples of 70 = 70, 140, 210, 280, 350, 420, 490, 560, 630, 700, . . . ., 6720, 6790, 6860, 6930, . . . .
- Multiples of 77 = 77, 154, 231, 308, 385, 462, 539, 616, 693, 770, . . . ., 6699, 6776, 6853, 6930, . . . .
Therefore, the LCM of 63, 70, and 77 is 6930.
Example 3: Calculate the LCM of 63, 70, and 77 using the GCD of the given numbers.

Solution:

Prime factorization of 63, 70, 77:
- 63 = 3² × 7¹
- 70 = 2¹ × 5¹ × 7¹
- 77 = 7¹ × 11¹
Therefore, GCD(63, 70) = 7, GCD(70, 77) = 7, GCD(63, 77) = 7, GCD(63, 70, 77) = 7
We know,
LCM(63, 70, 77) = [(63 × 70 × 77) × GCD(63, 70, 77)]/[GCD(63, 70) × GCD(70, 77) × GCD(63, 77)]
LCM(63, 70, 77) = (339570 × 7)/(7 × 7 × 7) = 6930
⇒LCM(63, 70, 77) = 6930

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FAQs on LCM of 63, 70, and 77

What is the LCM of 63, 70, and 77?

The LCM of 63, 70, and 77 is 6930. To find the least common multiple of 63, 70, and 77, we need to find the multiples of 63, 70, and 77 (multiples of 63 = 63, 126, 189, 252 . . . . 6930 . . . . ; multiples of 70 = 70, 140, 210, 280 . . . . 6930 . . . . ; multiples of 77 = 77, 154, 231, 308 . . . . 6930 . . . . ) and choose the smallest multiple that is exactly divisible by 63, 70, and 77, i.e., 6930.

What is the Least Perfect Square Divisible by 63, 70, and 77?

The least number divisible by 63, 70, and 77 = LCM(63, 70, 77)
LCM of 63, 70, and 77 = 2 × 3 × 3 × 5 × 7 × 11 [Incomplete pair(s): 2, 5, 7, 11]
⇒ Least perfect square divisible by each 63, 70, and 77 = LCM(63, 70, 77) × 2 × 5 × 7 × 11 = 5336100 [Square root of 5336100 = √5336100 = ±2310]
Therefore, 5336100 is the required number.

What is the Relation Between GCF and LCM of 63, 70, 77?

The following equation can be used to express the relation between GCF and LCM of 63, 70, 77, i.e. LCM(63, 70, 77) = [(63 × 70 × 77) × GCF(63, 70, 77)]/[GCF(63, 70) × GCF(70, 77) × GCF(63, 77)].

How to Find the LCM of 63, 70, and 77 by Prime Factorization?

To find the LCM of 63, 70, and 77 using prime factorization, we will find the prime factors, (63 = 3² × 7¹), (70 = 2¹ × 5¹ × 7¹), and (77 = 7¹ × 11¹). LCM of 63, 70, and 77 is the product of prime factors raised to their respective highest exponent among the numbers 63, 70, and 77.
⇒ LCM of 63, 70, 77 = 2¹ × 3² × 5¹ × 7¹ × 11¹ = 6930.

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