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LCM of 9, 10, and 28

LCM of 9, 10, and 28 is the smallest number among all common multiples of 9, 10, and 28. The first few multiples of 9, 10, and 28 are (9, 18, 27, 36, 45 . . .), (10, 20, 30, 40, 50 . . .), and (28, 56, 84, 112, 140 . . .) respectively. There are 3 commonly used methods to find LCM of 9, 10, 28 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 9, 10, and 28
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 9, 10, and 28?

Answer: LCM of 9, 10, and 28 is 1260.

Explanation:

The LCM of three non-zero integers, a(9), b(10), and c(28), is the smallest positive integer m(1260) that is divisible by a(9), b(10), and c(28) without any remainder.

Methods to Find LCM of 9, 10, and 28

Let's look at the different methods for finding the LCM of 9, 10, and 28.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 9, 10, and 28 by Division Method

To calculate the LCM of 9, 10, and 28 by the division method, we will divide the numbers(9, 10, 28) by their prime factors (preferably common). The product of these divisors gives the LCM of 9, 10, and 28.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 9, 10, and 28. Write this prime number(2) on the left of the given numbers(9, 10, and 28), separated as per the ladder arrangement.
Step 2: If any of the given numbers (9, 10, 28) 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 9, 10, and 28 is the product of all prime numbers on the left, i.e. LCM(9, 10, 28) by division method = 2 × 2 × 3 × 3 × 5 × 7 = 1260.

LCM of 9, 10, and 28 by Prime Factorization

Prime factorization of 9, 10, and 28 is (3 × 3) = 3², (2 × 5) = 2¹ × 5¹, and (2 × 2 × 7) = 2² × 7¹ respectively. LCM of 9, 10, and 28 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 5¹ × 7¹ = 1260.
Hence, the LCM of 9, 10, and 28 by prime factorization is 1260.

LCM of 9, 10, and 28 by Listing Multiples

To calculate the LCM of 9, 10, 28 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 9 (9, 18, 27, 36, 45 . . .), 10 (10, 20, 30, 40, 50 . . .), and 28 (28, 56, 84, 112, 140 . . .).
Step 2: The common multiples from the multiples of 9, 10, and 28 are 1260, 2520, . . .
Step 3: The smallest common multiple of 9, 10, and 28 is 1260.

∴ The least common multiple of 9, 10, and 28 = 1260.

☛ Also Check:

LCM of 9, 10, and 28 Examples

Example 1: Calculate the LCM of 9, 10, and 28 using the GCD of the given numbers.

Solution:

Prime factorization of 9, 10, 28:
- 9 = 3²
- 10 = 2¹ × 5¹
- 28 = 2² × 7¹
Therefore, GCD(9, 10) = 1, GCD(10, 28) = 2, GCD(9, 28) = 1, GCD(9, 10, 28) = 1
We know,
LCM(9, 10, 28) = [(9 × 10 × 28) × GCD(9, 10, 28)]/[GCD(9, 10) × GCD(10, 28) × GCD(9, 28)]
LCM(9, 10, 28) = (2520 × 1)/(1 × 2 × 1) = 1260
⇒LCM(9, 10, 28) = 1260
Example 2: Find the smallest number that is divisible by 9, 10, 28 exactly.

Solution:

The value of LCM(9, 10, 28) will be the smallest number that is exactly divisible by 9, 10, and 28.
⇒ Multiples of 9, 10, and 28:
- Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, . . . ., 1233, 1242, 1251, 1260, . . . .
- Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, . . . ., 1230, 1240, 1250, 1260, . . . .
- Multiples of 28 = 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, . . . ., 1204, 1232, 1260, . . . .
Therefore, the LCM of 9, 10, and 28 is 1260.
Example 3: Verify the relationship between the GCD and LCM of 9, 10, and 28.

Solution:

The relation between GCD and LCM of 9, 10, and 28 is given as,
LCM(9, 10, 28) = [(9 × 10 × 28) × GCD(9, 10, 28)]/[GCD(9, 10) × GCD(10, 28) × GCD(9, 28)]
⇒ Prime factorization of 9, 10 and 28:
- 9 = 3²
- 10 = 2¹ × 5¹
- 28 = 2² × 7¹
∴ GCD of (9, 10), (10, 28), (9, 28) and (9, 10, 28) = 1, 2, 1 and 1 respectively.
Now, LHS = LCM(9, 10, 28) = 1260.
And, RHS = [(9 × 10 × 28) × GCD(9, 10, 28)]/[GCD(9, 10) × GCD(10, 28) × GCD(9, 28)] = [(2520) × 1]/[1 × 2 × 1] = 1260
LHS = RHS = 1260.
Hence verified.

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FAQs on LCM of 9, 10, and 28

What is the LCM of 9, 10, and 28?

The LCM of 9, 10, and 28 is 1260. To find the LCM of 9, 10, and 28, we need to find the multiples of 9, 10, and 28 (multiples of 9 = 9, 18, 27, 36 . . . . 1260 . . . . ; multiples of 10 = 10, 20, 30, 40 . . . . 1260 . . . . ; multiples of 28 = 28, 56, 84, 112 . . . . 1260 . . . . ) and choose the smallest multiple that is exactly divisible by 9, 10, and 28, i.e., 1260.

How to Find the LCM of 9, 10, and 28 by Prime Factorization?

To find the LCM of 9, 10, and 28 using prime factorization, we will find the prime factors, (9 = 3²), (10 = 2¹ × 5¹), and (28 = 2² × 7¹). LCM of 9, 10, and 28 is the product of prime factors raised to their respective highest exponent among the numbers 9, 10, and 28.
⇒ LCM of 9, 10, 28 = 2² × 3² × 5¹ × 7¹ = 1260.

What is the Least Perfect Square Divisible by 9, 10, and 28?

The least number divisible by 9, 10, and 28 = LCM(9, 10, 28)
LCM of 9, 10, and 28 = 2 × 2 × 3 × 3 × 5 × 7 [Incomplete pair(s): 5, 7]
⇒ Least perfect square divisible by each 9, 10, and 28 = LCM(9, 10, 28) × 5 × 7 = 44100 [Square root of 44100 = √44100 = ±210]
Therefore, 44100 is the required number.

What is the Relation Between GCF and LCM of 9, 10, 28?

The following equation can be used to express the relation between GCF and LCM of 9, 10, 28, i.e. LCM(9, 10, 28) = [(9 × 10 × 28) × GCF(9, 10, 28)]/[GCF(9, 10) × GCF(10, 28) × GCF(9, 28)].

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