LCM of 19 and 21
LCM of 19 and 21 is the smallest number among all common multiples of 19 and 21. The first few multiples of 19 and 21 are (19, 38, 57, 76, 95, 114, 133, . . . ) and (21, 42, 63, 84, 105, . . . ) respectively. There are 3 commonly used methods to find LCM of 19 and 21 - by prime factorization, by listing multiples, and by division method.
1. | LCM of 19 and 21 |
2. | List of Methods |
3. | Solved Examples |
4. | FAQs |
What is the LCM of 19 and 21?
Answer: LCM of 19 and 21 is 399.
Explanation:
The LCM of two non-zero integers, x(19) and y(21), is the smallest positive integer m(399) that is divisible by both x(19) and y(21) without any remainder.
Methods to Find LCM of 19 and 21
Let's look at the different methods for finding the LCM of 19 and 21.
- By Prime Factorization Method
- By Division Method
- By Listing Multiples
LCM of 19 and 21 by Prime Factorization
Prime factorization of 19 and 21 is (19) = 191 and (3 × 7) = 31 × 71 respectively. LCM of 19 and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 31 × 71 × 191 = 399.
Hence, the LCM of 19 and 21 by prime factorization is 399.
LCM of 19 and 21 by Division Method
To calculate the LCM of 19 and 21 by the division method, we will divide the numbers(19, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 19 and 21.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 19 and 21. Write this prime number(3) on the left of the given numbers(19 and 21), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (19, 21) is a multiple of 3, divide it by 3 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 19 and 21 is the product of all prime numbers on the left, i.e. LCM(19, 21) by division method = 3 × 7 × 19 = 399.
LCM of 19 and 21 by Listing Multiples
To calculate the LCM of 19 and 21 by listing out the common multiples, we can follow the given below steps:
- Step 1: List a few multiples of 19 (19, 38, 57, 76, 95, 114, 133, . . . ) and 21 (21, 42, 63, 84, 105, . . . . )
- Step 2: The common multiples from the multiples of 19 and 21 are 399, 798, . . .
- Step 3: The smallest common multiple of 19 and 21 is 399.
∴ The least common multiple of 19 and 21 = 399.
☛ Also Check:
- LCM of 9 and 12 - 36
- LCM of 6, 72 and 120 - 360
- LCM of 48 and 64 - 192
- LCM of 8 and 9 - 72
- LCM of 9 and 45 - 45
- LCM of 50 and 75 - 150
- LCM of 3, 5 and 15 - 15
LCM of 19 and 21 Examples
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Example 1: Find the smallest number that is divisible by 19 and 21 exactly.
Solution:
The value of LCM(19, 21) will be the smallest number that is exactly divisible by 19 and 21.
⇒ Multiples of 19 and 21:- Multiples of 19 = 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, . . . ., 323, 342, 361, 380, 399, . . . .
- Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 357, 378, 399, . . . .
Therefore, the LCM of 19 and 21 is 399.
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Example 2: The GCD and LCM of two numbers are 1 and 399 respectively. If one number is 19, find the other number.
Solution:
Let the other number be b.
∵ GCD × LCM = 19 × b
⇒ b = (GCD × LCM)/19
⇒ b = (1 × 399)/19
⇒ b = 21
Therefore, the other number is 21. -
Example 3: Verify the relationship between GCF and LCM of 19 and 21.
Solution:
The relation between GCF and LCM of 19 and 21 is given as,
LCM(19, 21) × GCF(19, 21) = Product of 19, 21
Prime factorization of 19 and 21 is given as, 19 = (19) = 191 and 21 = (3 × 7) = 31 × 71
LCM(19, 21) = 399
GCF(19, 21) = 1
LHS = LCM(19, 21) × GCF(19, 21) = 399 × 1 = 399
RHS = Product of 19, 21 = 19 × 21 = 399
⇒ LHS = RHS = 399
Hence, verified.
FAQs on LCM of 19 and 21
What is the LCM of 19 and 21?
The LCM of 19 and 21 is 399. To find the least common multiple (LCM) of 19 and 21, we need to find the multiples of 19 and 21 (multiples of 19 = 19, 38, 57, 76 . . . . 399; multiples of 21 = 21, 42, 63, 84 . . . . 399) and choose the smallest multiple that is exactly divisible by 19 and 21, i.e., 399.
How to Find the LCM of 19 and 21 by Prime Factorization?
To find the LCM of 19 and 21 using prime factorization, we will find the prime factors, (19 = 19) and (21 = 3 × 7). LCM of 19 and 21 is the product of prime factors raised to their respective highest exponent among the numbers 19 and 21.
⇒ LCM of 19, 21 = 31 × 71 × 191 = 399.
What is the Relation Between GCF and LCM of 19, 21?
The following equation can be used to express the relation between GCF and LCM of 19 and 21, i.e. GCF × LCM = 19 × 21.
If the LCM of 21 and 19 is 399, Find its GCF.
LCM(21, 19) × GCF(21, 19) = 21 × 19
Since the LCM of 21 and 19 = 399
⇒ 399 × GCF(21, 19) = 399
Therefore, the greatest common factor = 399/399 = 1.
What are the Methods to Find LCM of 19 and 21?
The commonly used methods to find the LCM of 19 and 21 are:
- Division Method
- Listing Multiples
- Prime Factorization Method
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