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LCM of 7, 11, 21, and 22

LCM of 7, 11, 21, and 22 is the smallest number among all common multiples of 7, 11, 21, and 22. The first few multiples of 7, 11, 21, and 22 are (7, 14, 21, 28, 35 . . .), (11, 22, 33, 44, 55 . . .), (21, 42, 63, 84, 105 . . .), and (22, 44, 66, 88, 110 . . .) respectively. There are 3 commonly used methods to find LCM of 7, 11, 21, 22 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 7, 11, 21, and 22
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 7, 11, 21, and 22?

Answer: LCM of 7, 11, 21, and 22 is 462.

Explanation:

The LCM of four non-zero integers, a(7), b(11), c(21), and d(22), is the smallest positive integer m(462) that is divisible by a(7), b(11), c(21), and d(22) without any remainder.

Methods to Find LCM of 7, 11, 21, and 22

The methods to find the LCM of 7, 11, 21, and 22 are explained below.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 7, 11, 21, and 22 by Prime Factorization

Prime factorization of 7, 11, 21, and 22 is (7) = 7¹, (11) = 11¹, (3 × 7) = 3¹ × 7¹, and (2 × 11) = 2¹ × 11¹ respectively. LCM of 7, 11, 21, and 22 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3¹ × 7¹ × 11¹ = 462.
Hence, the LCM of 7, 11, 21, and 22 by prime factorization is 462.

LCM of 7, 11, 21, and 22 by Division Method

To calculate the LCM of 7, 11, 21, and 22 by the division method, we will divide the numbers(7, 11, 21, 22) by their prime factors (preferably common). The product of these divisors gives the LCM of 7, 11, 21, and 22.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 7, 11, 21, and 22. Write this prime number(2) on the left of the given numbers(7, 11, 21, and 22), separated as per the ladder arrangement.
Step 2: If any of the given numbers (7, 11, 21, 22) 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 7, 11, 21, and 22 is the product of all prime numbers on the left, i.e. LCM(7, 11, 21, 22) by division method = 2 × 3 × 7 × 11 = 462.

LCM of 7, 11, 21, and 22 by Listing Multiples

To calculate the LCM of 7, 11, 21, 22 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 7 (7, 14, 21, 28, 35 . . .), 11 (11, 22, 33, 44, 55 . . .), 21 (21, 42, 63, 84, 105 . . .), and 22 (22, 44, 66, 88, 110 . . .).
Step 2: The common multiples from the multiples of 7, 11, 21, and 22 are 462, 924, . . .
Step 3: The smallest common multiple of 7, 11, 21, and 22 is 462.

∴ The least common multiple of 7, 11, 21, and 22 = 462.

☛ Also Check:

LCM of 7, 11, 21, and 22 Examples

Example 1: Find the smallest number which when divided by 7, 11, 21, and 22 leaves 5 as the remainder in each case.

Solution:

The smallest number exactly divisible by 7, 11, 21, and 22 = LCM(7, 11, 21, 22) ⇒ Smallest number which leaves 5 as remainder when divided by 7, 11, 21, and 22 = LCM(7, 11, 21, 22) + 5
- 7 = 7¹
- 11 = 11¹
- 21 = 3¹ × 7¹
- 22 = 2¹ × 11¹
LCM(7, 11, 21, 22) = 2¹ × 3¹ × 7¹ × 11¹ = 462
⇒ The required number = 462 + 5 = 467.
Example 2: Find the smallest number that is divisible by 7, 11, 21, 22 exactly.

Solution:

The value of LCM(7, 11, 21, 22) will be the smallest number that is exactly divisible by 7, 11, 21, and 22.
⇒ Multiples of 7, 11, 21, and 22:
- Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, . . . ., 441, 448, 455, 462, . . . .
- Multiples of 11 = 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, . . . ., 418, 429, 440, 451, 462, . . . .
- Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 399, 420, 441, 462, . . . .
- Multiples of 22 = 22, 44, 66, 88, 110, 132, 154, 176, 198, 220, . . . ., 418, 440, 462, . . . .
Therefore, the LCM of 7, 11, 21, and 22 is 462.
Example 3: Which of the following is the LCM of 7, 11, 21, 22? 462, 11, 35, 120.

Solution:

The value of LCM of 7, 11, 21, and 22 is the smallest common multiple of 7, 11, 21, and 22. The number satisfying the given condition is 462. ∴LCM(7, 11, 21, 22) = 462.

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FAQs on LCM of 7, 11, 21, and 22

What is the LCM of 7, 11, 21, and 22?

The LCM of 7, 11, 21, and 22 is 462. To find the least common multiple (LCM) of 7, 11, 21, and 22, we need to find the multiples of 7, 11, 21, and 22 (multiples of 7 = 7, 14, 21, 28 . . . . 462 . . . . ; multiples of 11 = 11, 22, 33, 44 . . . . 462 . . . . ; multiples of 21 = 21, 42, 63, 84 . . . . 462 . . . . ; multiples of 22 = 22, 44, 66, 88 . . . . 462 . . . . ) and choose the smallest multiple that is exactly divisible by 7, 11, 21, and 22, i.e., 462.

How to Find the LCM of 7, 11, 21, and 22 by Prime Factorization?

To find the LCM of 7, 11, 21, and 22 using prime factorization, we will find the prime factors, (7 = 7¹), (11 = 11¹), (21 = 3¹ × 7¹), and (22 = 2¹ × 11¹). LCM of 7, 11, 21, and 22 is the product of prime factors raised to their respective highest exponent among the numbers 7, 11, 21, and 22.
⇒ LCM of 7, 11, 21, 22 = 2¹ × 3¹ × 7¹ × 11¹ = 462.

What are the Methods to Find LCM of 7, 11, 21, 22?

The commonly used methods to find the LCM of 7, 11, 21, 22 are:

Prime Factorization Method
Division Method
Listing Multiples

Which of the following is the LCM of 7, 11, 21, and 22? 24, 32, 10, 462

The value of LCM of 7, 11, 21, 22 is the smallest common multiple of 7, 11, 21, and 22. The number satisfying the given condition is 462.

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