LCM of 5, 15, and 20
LCM of 5, 15, and 20 is the smallest number among all common multiples of 5, 15, and 20. The first few multiples of 5, 15, and 20 are (5, 10, 15, 20, 25 . . .), (15, 30, 45, 60, 75 . . .), and (20, 40, 60, 80, 100 . . .) respectively. There are 3 commonly used methods to find LCM of 5, 15, 20 - by division method, by listing multiples, and by prime factorization.
1. | LCM of 5, 15, and 20 |
2. | List of Methods |
3. | Solved Examples |
4. | FAQs |
What is the LCM of 5, 15, and 20?
Answer: LCM of 5, 15, and 20 is 60.
Explanation:
The LCM of three non-zero integers, a(5), b(15), and c(20), is the smallest positive integer m(60) that is divisible by a(5), b(15), and c(20) without any remainder.
Methods to Find LCM of 5, 15, and 20
Let's look at the different methods for finding the LCM of 5, 15, and 20.
- By Division Method
- By Listing Multiples
- By Prime Factorization Method
LCM of 5, 15, and 20 by Division Method
To calculate the LCM of 5, 15, and 20 by the division method, we will divide the numbers(5, 15, 20) by their prime factors (preferably common). The product of these divisors gives the LCM of 5, 15, and 20.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 5, 15, and 20. Write this prime number(2) on the left of the given numbers(5, 15, and 20), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (5, 15, 20) 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 5, 15, and 20 is the product of all prime numbers on the left, i.e. LCM(5, 15, 20) by division method = 2 × 2 × 3 × 5 = 60.
LCM of 5, 15, and 20 by Listing Multiples
To calculate the LCM of 5, 15, 20 by listing out the common multiples, we can follow the given below steps:
- Step 1: List a few multiples of 5 (5, 10, 15, 20, 25 . . .), 15 (15, 30, 45, 60, 75 . . .), and 20 (20, 40, 60, 80, 100 . . .).
- Step 2: The common multiples from the multiples of 5, 15, and 20 are 60, 120, . . .
- Step 3: The smallest common multiple of 5, 15, and 20 is 60.
∴ The least common multiple of 5, 15, and 20 = 60.
LCM of 5, 15, and 20 by Prime Factorization
Prime factorization of 5, 15, and 20 is (5) = 51, (3 × 5) = 31 × 51, and (2 × 2 × 5) = 22 × 51 respectively. LCM of 5, 15, and 20 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 31 × 51 = 60.
Hence, the LCM of 5, 15, and 20 by prime factorization is 60.
☛ Also Check:
- LCM of 11 and 22 - 22
- LCM of 16, 24, 36 and 54 - 432
- LCM of 2, 4 and 7 - 28
- LCM of 2, 4 and 6 - 12
- LCM of 3 and 13 - 39
- LCM of 63, 70 and 77 - 6930
- LCM of 30 and 40 - 120
LCM of 5, 15, and 20 Examples
-
Example 1: Verify the relationship between the GCD and LCM of 5, 15, and 20.
Solution:
The relation between GCD and LCM of 5, 15, and 20 is given as,
LCM(5, 15, 20) = [(5 × 15 × 20) × GCD(5, 15, 20)]/[GCD(5, 15) × GCD(15, 20) × GCD(5, 20)]
⇒ Prime factorization of 5, 15 and 20:- 5 = 51
- 15 = 31 × 51
- 20 = 22 × 51
∴ GCD of (5, 15), (15, 20), (5, 20) and (5, 15, 20) = 5, 5, 5 and 5 respectively.
Now, LHS = LCM(5, 15, 20) = 60.
And, RHS = [(5 × 15 × 20) × GCD(5, 15, 20)]/[GCD(5, 15) × GCD(15, 20) × GCD(5, 20)] = [(1500) × 5]/[5 × 5 × 5] = 60
LHS = RHS = 60.
Hence verified. -
Example 2: Find the smallest number that is divisible by 5, 15, 20 exactly.
Solution:
The smallest number that is divisible by 5, 15, and 20 exactly is their LCM.
⇒ Multiples of 5, 15, and 20:- Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, . . . .
- Multiples of 20 = 20, 40, 60, 80, 100, . . . .
Therefore, the LCM of 5, 15, and 20 is 60.
-
Example 3: Calculate the LCM of 5, 15, and 20 using the GCD of the given numbers.
Solution:
Prime factorization of 5, 15, 20:
- 5 = 51
- 15 = 31 × 51
- 20 = 22 × 51
Therefore, GCD(5, 15) = 5, GCD(15, 20) = 5, GCD(5, 20) = 5, GCD(5, 15, 20) = 5
We know,
LCM(5, 15, 20) = [(5 × 15 × 20) × GCD(5, 15, 20)]/[GCD(5, 15) × GCD(15, 20) × GCD(5, 20)]
LCM(5, 15, 20) = (1500 × 5)/(5 × 5 × 5) = 60
⇒LCM(5, 15, 20) = 60
FAQs on LCM of 5, 15, and 20
What is the LCM of 5, 15, and 20?
The LCM of 5, 15, and 20 is 60. To find the least common multiple (LCM) of 5, 15, and 20, we need to find the multiples of 5, 15, and 20 (multiples of 5 = 5, 10, 15, 20 . . . . 60 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . .; multiples of 20 = 20, 40, 60, 80 . . . .) and choose the smallest multiple that is exactly divisible by 5, 15, and 20, i.e., 60.
What are the Methods to Find LCM of 5, 15, 20?
The commonly used methods to find the LCM of 5, 15, 20 are:
- Prime Factorization Method
- Listing Multiples
- Division Method
How to Find the LCM of 5, 15, and 20 by Prime Factorization?
To find the LCM of 5, 15, and 20 using prime factorization, we will find the prime factors, (5 = 51), (15 = 31 × 51), and (20 = 22 × 51). LCM of 5, 15, and 20 is the product of prime factors raised to their respective highest exponent among the numbers 5, 15, and 20.
⇒ LCM of 5, 15, 20 = 22 × 31 × 51 = 60.
Which of the following is the LCM of 5, 15, and 20? 24, 60, 120, 32
The value of LCM of 5, 15, 20 is the smallest common multiple of 5, 15, and 20. The number satisfying the given condition is 60.
visual curriculum