LCM of 60 and 84
LCM of 60 and 84 is the smallest number among all common multiples of 60 and 84. The first few multiples of 60 and 84 are (60, 120, 180, 240, 300, . . . ) and (84, 168, 252, 336, 420, 504, . . . ) respectively. There are 3 commonly used methods to find LCM of 60 and 84  by listing multiples, by prime factorization, and by division method.
1.  LCM of 60 and 84 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 60 and 84?
Answer: LCM of 60 and 84 is 420.
Explanation:
The LCM of two nonzero integers, x(60) and y(84), is the smallest positive integer m(420) that is divisible by both x(60) and y(84) without any remainder.
Methods to Find LCM of 60 and 84
Let's look at the different methods for finding the LCM of 60 and 84.
 By Listing Multiples
 By Prime Factorization Method
 By Division Method
LCM of 60 and 84 by Listing Multiples
To calculate the LCM of 60 and 84 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 60 (60, 120, 180, 240, 300, . . . ) and 84 (84, 168, 252, 336, 420, 504, . . . . )
 Step 2: The common multiples from the multiples of 60 and 84 are 420, 840, . . .
 Step 3: The smallest common multiple of 60 and 84 is 420.
∴ The least common multiple of 60 and 84 = 420.
LCM of 60 and 84 by Prime Factorization
Prime factorization of 60 and 84 is (2 × 2 × 3 × 5) = 2^{2} × 3^{1} × 5^{1} and (2 × 2 × 3 × 7) = 2^{2} × 3^{1} × 7^{1} respectively. LCM of 60 and 84 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{2} × 3^{1} × 5^{1} × 7^{1} = 420.
Hence, the LCM of 60 and 84 by prime factorization is 420.
LCM of 60 and 84 by Division Method
To calculate the LCM of 60 and 84 by the division method, we will divide the numbers(60, 84) by their prime factors (preferably common). The product of these divisors gives the LCM of 60 and 84.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 60 and 84. Write this prime number(2) on the left of the given numbers(60 and 84), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (60, 84) 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 60 and 84 is the product of all prime numbers on the left, i.e. LCM(60, 84) by division method = 2 × 2 × 3 × 5 × 7 = 420.
☛ Also Check:
 LCM of 4, 10 and 12  60
 LCM of 24 and 84  168
 LCM of 3, 4 and 6  12
 LCM of 10, 25, 35 and 40  1400
 LCM of 25 and 65  325
 LCM of 70, 105 and 175  1050
 LCM of 16 and 64  64
LCM of 60 and 84 Examples

Example 1: Verify the relationship between GCF and LCM of 60 and 84.
Solution:
The relation between GCF and LCM of 60 and 84 is given as,
LCM(60, 84) × GCF(60, 84) = Product of 60, 84
Prime factorization of 60 and 84 is given as, 60 = (2 × 2 × 3 × 5) = 2^{2} × 3^{1} × 5^{1} and 84 = (2 × 2 × 3 × 7) = 2^{2} × 3^{1} × 7^{1}
LCM(60, 84) = 420
GCF(60, 84) = 12
LHS = LCM(60, 84) × GCF(60, 84) = 420 × 12 = 5040
RHS = Product of 60, 84 = 60 × 84 = 5040
⇒ LHS = RHS = 5040
Hence, verified. 
Example 2: The GCD and LCM of two numbers are 12 and 420 respectively. If one number is 84, find the other number.
Solution:
Let the other number be a.
∵ GCD × LCM = 84 × a
⇒ a = (GCD × LCM)/84
⇒ a = (12 × 420)/84
⇒ a = 60
Therefore, the other number is 60. 
Example 3: Find the smallest number that is divisible by 60 and 84 exactly.
Solution:
The smallest number that is divisible by 60 and 84 exactly is their LCM.
⇒ Multiples of 60 and 84: Multiples of 60 = 60, 120, 180, 240, 300, 360, 420, . . . .
 Multiples of 84 = 84, 168, 252, 336, 420, . . . .
Therefore, the LCM of 60 and 84 is 420.
FAQs on LCM of 60 and 84
What is the LCM of 60 and 84?
The LCM of 60 and 84 is 420. To find the LCM of 60 and 84, we need to find the multiples of 60 and 84 (multiples of 60 = 60, 120, 180, 240 . . . . 420; multiples of 84 = 84, 168, 252, 336 . . . . 420) and choose the smallest multiple that is exactly divisible by 60 and 84, i.e., 420.
If the LCM of 84 and 60 is 420, Find its GCF.
LCM(84, 60) × GCF(84, 60) = 84 × 60
Since the LCM of 84 and 60 = 420
⇒ 420 × GCF(84, 60) = 5040
Therefore, the GCF (greatest common factor) = 5040/420 = 12.
Which of the following is the LCM of 60 and 84? 35, 2, 420, 30
The value of LCM of 60, 84 is the smallest common multiple of 60 and 84. The number satisfying the given condition is 420.
What is the Relation Between GCF and LCM of 60, 84?
The following equation can be used to express the relation between GCF and LCM of 60 and 84, i.e. GCF × LCM = 60 × 84.
What is the Least Perfect Square Divisible by 60 and 84?
The least number divisible by 60 and 84 = LCM(60, 84)
LCM of 60 and 84 = 2 × 2 × 3 × 5 × 7 [Incomplete pair(s): 3, 5, 7]
⇒ Least perfect square divisible by each 60 and 84 = LCM(60, 84) × 3 × 5 × 7 = 44100 [Square root of 44100 = √44100 = ±210]
Therefore, 44100 is the required number.
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