LCM of 8 and 42
LCM of 8 and 42 is the smallest number among all common multiples of 8 and 42. The first few multiples of 8 and 42 are (8, 16, 24, 32, 40, 48, . . . ) and (42, 84, 126, 168, 210, 252, 294, . . . ) respectively. There are 3 commonly used methods to find LCM of 8 and 42 - by listing multiples, by division method, and by prime factorization.
1. | LCM of 8 and 42 |
2. | List of Methods |
3. | Solved Examples |
4. | FAQs |
What is the LCM of 8 and 42?
Answer: LCM of 8 and 42 is 168.
![LCM of 8 and 42](https://static.qumath.in/static/website/old-cdn-static/common-factor/lcm-of-8-and-42.png)
Explanation:
The LCM of two non-zero integers, x(8) and y(42), is the smallest positive integer m(168) that is divisible by both x(8) and y(42) without any remainder.
Methods to Find LCM of 8 and 42
Let's look at the different methods for finding the LCM of 8 and 42.
- By Listing Multiples
- By Prime Factorization Method
- By Division Method
LCM of 8 and 42 by Listing Multiples
![LCM of 8 and 42 by Listing Multiples Method](https://static.qumath.in/static/website/old-cdn-static/common-factor/lcm-of-8-and-42-by-listing-multiples-method.png)
To calculate the LCM of 8 and 42 by listing out the common multiples, we can follow the given below steps:
- Step 1: List a few multiples of 8 (8, 16, 24, 32, 40, 48, . . . ) and 42 (42, 84, 126, 168, 210, 252, 294, . . . . )
- Step 2: The common multiples from the multiples of 8 and 42 are 168, 336, . . .
- Step 3: The smallest common multiple of 8 and 42 is 168.
∴ The least common multiple of 8 and 42 = 168.
LCM of 8 and 42 by Prime Factorization
Prime factorization of 8 and 42 is (2 × 2 × 2) = 23 and (2 × 3 × 7) = 21 × 31 × 71 respectively. LCM of 8 and 42 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 23 × 31 × 71 = 168.
Hence, the LCM of 8 and 42 by prime factorization is 168.
LCM of 8 and 42 by Division Method
To calculate the LCM of 8 and 42 by the division method, we will divide the numbers(8, 42) by their prime factors (preferably common). The product of these divisors gives the LCM of 8 and 42.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8 and 42. Write this prime number(2) on the left of the given numbers(8 and 42), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (8, 42) 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 8 and 42 is the product of all prime numbers on the left, i.e. LCM(8, 42) by division method = 2 × 2 × 2 × 3 × 7 = 168.
☛ Also Check:
- LCM of 10 and 25 - 50
- LCM of 10 and 16 - 80
- LCM of 2, 5 and 6 - 30
- LCM of 36 and 81 - 324
- LCM of 40, 48 and 45 - 720
- LCM of 42 and 70 - 210
- LCM of 7 and 10 - 70
LCM of 8 and 42 Examples
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Example 1: Verify the relationship between GCF and LCM of 8 and 42.
Solution:
The relation between GCF and LCM of 8 and 42 is given as,
LCM(8, 42) × GCF(8, 42) = Product of 8, 42
Prime factorization of 8 and 42 is given as, 8 = (2 × 2 × 2) = 23 and 42 = (2 × 3 × 7) = 21 × 31 × 71
LCM(8, 42) = 168
GCF(8, 42) = 2
LHS = LCM(8, 42) × GCF(8, 42) = 168 × 2 = 336
RHS = Product of 8, 42 = 8 × 42 = 336
⇒ LHS = RHS = 336
Hence, verified. -
Example 2: Find the smallest number that is divisible by 8 and 42 exactly.
Solution:
The value of LCM(8, 42) will be the smallest number that is exactly divisible by 8 and 42.
⇒ Multiples of 8 and 42:- Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, . . . ., 136, 144, 152, 160, 168, . . . .
- Multiples of 42 = 42, 84, 126, 168, 210, 252, 294, 336, 378, 420, . . . ., 0, 42, 84, 126, 168, . . . .
Therefore, the LCM of 8 and 42 is 168.
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Example 3: The product of two numbers is 336. If their GCD is 2, what is their LCM?
Solution:
Given: GCD = 2
product of numbers = 336
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 336/2
Therefore, the LCM is 168.
The probable combination for the given case is LCM(8, 42) = 168.
FAQs on LCM of 8 and 42
What is the LCM of 8 and 42?
The LCM of 8 and 42 is 168. To find the least common multiple (LCM) of 8 and 42, we need to find the multiples of 8 and 42 (multiples of 8 = 8, 16, 24, 32 . . . . 168; multiples of 42 = 42, 84, 126, 168) and choose the smallest multiple that is exactly divisible by 8 and 42, i.e., 168.
How to Find the LCM of 8 and 42 by Prime Factorization?
To find the LCM of 8 and 42 using prime factorization, we will find the prime factors, (8 = 2 × 2 × 2) and (42 = 2 × 3 × 7). LCM of 8 and 42 is the product of prime factors raised to their respective highest exponent among the numbers 8 and 42.
⇒ LCM of 8, 42 = 23 × 31 × 71 = 168.
What is the Least Perfect Square Divisible by 8 and 42?
The least number divisible by 8 and 42 = LCM(8, 42)
LCM of 8 and 42 = 2 × 2 × 2 × 3 × 7 [Incomplete pair(s): 2, 3, 7]
⇒ Least perfect square divisible by each 8 and 42 = LCM(8, 42) × 2 × 3 × 7 = 7056 [Square root of 7056 = √7056 = ±84]
Therefore, 7056 is the required number.
If the LCM of 42 and 8 is 168, Find its GCF.
LCM(42, 8) × GCF(42, 8) = 42 × 8
Since the LCM of 42 and 8 = 168
⇒ 168 × GCF(42, 8) = 336
Therefore, the greatest common factor (GCF) = 336/168 = 2.
What is the Relation Between GCF and LCM of 8, 42?
The following equation can be used to express the relation between GCF and LCM of 8 and 42, i.e. GCF × LCM = 8 × 42.
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