LCM of 6 and 7
LCM of 6 and 7 is the smallest number among all common multiples of 6 and 7. The first few multiples of 6 and 7 are (6, 12, 18, 24, 30, 36, . . . ) and (7, 14, 21, 28, 35, 42, 49, . . . ) respectively. There are 3 commonly used methods to find LCM of 6 and 7 - by division method, by prime factorization, and by listing multiples.
1. | LCM of 6 and 7 |
2. | List of Methods |
3. | Solved Examples |
4. | FAQs |
What is the LCM of 6 and 7?
Answer: LCM of 6 and 7 is 42.
Explanation:
The LCM of two non-zero integers, x(6) and y(7), is the smallest positive integer m(42) that is divisible by both x(6) and y(7) without any remainder.
Methods to Find LCM of 6 and 7
The methods to find the LCM of 6 and 7 are explained below.
- By Division Method
- By Listing Multiples
- By Prime Factorization Method
LCM of 6 and 7 by Division Method
To calculate the LCM of 6 and 7 by the division method, we will divide the numbers(6, 7) by their prime factors (preferably common). The product of these divisors gives the LCM of 6 and 7.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6 and 7. Write this prime number(2) on the left of the given numbers(6 and 7), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (6, 7) 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 6 and 7 is the product of all prime numbers on the left, i.e. LCM(6, 7) by division method = 2 × 3 × 7 = 42.
LCM of 6 and 7 by Listing Multiples
To calculate the LCM of 6 and 7 by listing out the common multiples, we can follow the given below steps:
- Step 1: List a few multiples of 6 (6, 12, 18, 24, 30, 36, . . . ) and 7 (7, 14, 21, 28, 35, 42, 49, . . . . )
- Step 2: The common multiples from the multiples of 6 and 7 are 42, 84, . . .
- Step 3: The smallest common multiple of 6 and 7 is 42.
∴ The least common multiple of 6 and 7 = 42.
LCM of 6 and 7 by Prime Factorization
Prime factorization of 6 and 7 is (2 × 3) = 21 × 31 and (7) = 71 respectively. LCM of 6 and 7 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 21 × 31 × 71 = 42.
Hence, the LCM of 6 and 7 by prime factorization is 42.
☛ Also Check:
- LCM of 4 and 5 - 20
- LCM of 8, 16 and 24 - 48
- LCM of 5, 8 and 12 - 120
- LCM of 24, 30 and 40 - 120
- LCM of 10 and 11 - 110
- LCM of 36 and 48 - 144
- LCM of 60 and 90 - 180
LCM of 6 and 7 Examples
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Example 1: Find the smallest number that is divisible by 6 and 7 exactly.
Solution:
The smallest number that is divisible by 6 and 7 exactly is their LCM.
⇒ Multiples of 6 and 7:- Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, . . . .
- Multiples of 7 = 7, 14, 21, 28, 35, 42, . . . .
Therefore, the LCM of 6 and 7 is 42.
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Example 2: Verify the relationship between GCF and LCM of 6 and 7.
Solution:
The relation between GCF and LCM of 6 and 7 is given as,
LCM(6, 7) × GCF(6, 7) = Product of 6, 7
Prime factorization of 6 and 7 is given as, 6 = (2 × 3) = 21 × 31 and 7 = (7) = 71
LCM(6, 7) = 42
GCF(6, 7) = 1
LHS = LCM(6, 7) × GCF(6, 7) = 42 × 1 = 42
RHS = Product of 6, 7 = 6 × 7 = 42
⇒ LHS = RHS = 42
Hence, verified. -
Example 3: The product of two numbers is 42. If their GCD is 1, what is their LCM?
Solution:
Given: GCD = 1
product of numbers = 42
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 42/1
Therefore, the LCM is 42.
The probable combination for the given case is LCM(6, 7) = 42.
FAQs on LCM of 6 and 7
What is the LCM of 6 and 7?
The LCM of 6 and 7 is 42. To find the least common multiple of 6 and 7, we need to find the multiples of 6 and 7 (multiples of 6 = 6, 12, 18, 24 . . . . 42; multiples of 7 = 7, 14, 21, 28 . . . . 42) and choose the smallest multiple that is exactly divisible by 6 and 7, i.e., 42.
What are the Methods to Find LCM of 6 and 7?
The commonly used methods to find the LCM of 6 and 7 are:
- Listing Multiples
- Division Method
- Prime Factorization Method
How to Find the LCM of 6 and 7 by Prime Factorization?
To find the LCM of 6 and 7 using prime factorization, we will find the prime factors, (6 = 2 × 3) and (7 = 7). LCM of 6 and 7 is the product of prime factors raised to their respective highest exponent among the numbers 6 and 7.
⇒ LCM of 6, 7 = 21 × 31 × 71 = 42.
What is the Relation Between GCF and LCM of 6, 7?
The following equation can be used to express the relation between GCF and LCM of 6 and 7, i.e. GCF × LCM = 6 × 7.
If the LCM of 7 and 6 is 42, Find its GCF.
LCM(7, 6) × GCF(7, 6) = 7 × 6
Since the LCM of 7 and 6 = 42
⇒ 42 × GCF(7, 6) = 42
Therefore, the GCF = 42/42 = 1.
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