The HCF and lcm of two numbers are 16 and 192 respectively one of the numbers is 48 find the other

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LCM of 16 and 48 is the smallest number among all common multiples of 16 and 48. The first few multiples of 16 and 48 are (16, 32, 48, 64, 80, 96, . . . ) and (48, 96, 144, 192, 240, 288, 336, . . . ) respectively. There are 3 commonly used methods to find LCM of 16 and 48 - by prime factorization, by division method, and by listing multiples.

What is the LCM of 16 and 48?

Answer: LCM of 16 and 48 is 48.

Explanation:

The LCM of two non-zero integers, x(16) and y(48), is the smallest positive integer m(48) that is divisible by both x(16) and y(48) without any remainder.

Methods to Find LCM of 16 and 48

Let's look at the different methods for finding the LCM of 16 and 48.

• By Division Method
• By Listing Multiples
• By Prime Factorization Method

LCM of 16 and 48 by Division Method

To calculate the LCM of 16 and 48 by the division method, we will divide the numbers(16, 48) by their prime factors (preferably common). The product of these divisors gives the LCM of 16 and 48.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 16 and 48. Write this prime number(2) on the left of the given numbers(16 and 48), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (16, 48) 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 16 and 48 is the product of all prime numbers on the left, i.e. LCM(16, 48) by division method = 2 × 2 × 2 × 2 × 3 = 48.

LCM of 16 and 48 by Listing Multiples

To calculate the LCM of 16 and 48 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 16 (16, 32, 48, 64, 80, 96, . . . ) and 48 (48, 96, 144, 192, 240, 288, 336, . . . . )
• Step 2: The common multiples from the multiples of 16 and 48 are 48, 96, . . .
• Step 3: The smallest common multiple of 16 and 48 is 48.

∴ The least common multiple of 16 and 48 = 48.

LCM of 16 and 48 by Prime Factorization

Prime factorization of 16 and 48 is (2 × 2 × 2 × 2) = 24 and (2 × 2 × 2 × 2 × 3) = 24 × 31 respectively. LCM of 16 and 48 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 24 × 31 = 48.
Hence, the LCM of 16 and 48 by prime factorization is 48.

☛ Also Check:

LCM of 16 and 48 Examples

1. Example 1: Find the smallest number that is divisible by 16 and 48 exactly.

   Solution:

   The smallest number that is divisible by 16 and 48 exactly is their LCM.
   ⇒ Multiples of 16 and 48:

   • Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, . . . .
   • Multiples of 48 = 48, 96, 144, 192, 240, 288, 336, . . . .

   Therefore, the LCM of 16 and 48 is 48.

2. Example 2: Verify the relationship between GCF and LCM of 16 and 48.

   Solution:

   The relation between GCF and LCM of 16 and 48 is given as, LCM(16, 48) × GCF(16, 48) = Product of 16, 48

   Prime factorization of 16 and 48 is given as, 16 = (2 × 2 × 2 × 2) = 24 and 48 = (2 × 2 × 2 × 2 × 3) = 24 × 31

   LCM(16, 48) = 48 GCF(16, 48) = 16 LHS = LCM(16, 48) × GCF(16, 48) = 48 × 16 = 768 RHS = Product of 16, 48 = 16 × 48 = 768 ⇒ LHS = RHS = 768

   Hence, verified.

Example 3: The GCD and LCM of two numbers are 16 and 48 respectively. If one number is 16, find the other number.

Solution:

Let the other number be m.
∵ GCD × LCM = 16 × m ⇒ m = (GCD × LCM)/16 ⇒ m = (16 × 48)/16 ⇒ m = 48

Therefore, the other number is 48.

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The LCM of 16 and 48 is 48. To find the LCM of 16 and 48, we need to find the multiples of 16 and 48 (multiples of 16 = 16, 32, 48, 64; multiples of 48 = 48, 96, 144, 192) and choose the smallest multiple that is exactly divisible by 16 and 48, i.e., 48.

Which of the following is the LCM of 16 and 48? 32, 25, 10, 48

The value of LCM of 16, 48 is the smallest common multiple of 16 and 48. The number satisfying the given condition is 48.

What are the Methods to Find LCM of 16 and 48?

The commonly used methods to find the LCM of 16 and 48 are:

• Listing Multiples
• Prime Factorization Method
• Division Method

If the LCM of 48 and 16 is 48, Find its GCF.

LCM(48, 16) × GCF(48, 16) = 48 × 16 Since the LCM of 48 and 16 = 48 ⇒ 48 × GCF(48, 16) = 768

Therefore, the greatest common factor (GCF) = 768/48 = 16.

How to Find the LCM of 16 and 48 by Prime Factorization?

To find the LCM of 16 and 48 using prime factorization, we will find the prime factors, (16 = 2 × 2 × 2 × 2) and (48 = 2 × 2 × 2 × 2 × 3). LCM of 16 and 48 is the product of prime factors raised to their respective highest exponent among the numbers 16 and 48.
⇒ LCM of 16, 48 = 24 × 31 = 48.

The HCF and LCM of two numbers is 16 and 192 respectively If one of the numbers is 64 the other one is We know that ,

Product of two numbers = LCM $$\displaystyle \times $$ HCF

 $$ x \displaystyle \times $$ 64 = 192 $$\displaystyle \times $$ 16

$$x=\displaystyle \frac{192\times 16}{64}=48$$
$$\displaystyle \therefore $$ other number $$= 48$$

Last updated - April 26, 2022

Answer: HCF = 48 and LCM = 192

Step by step solution:

Contents:

Given numbers = 48 and 192

To find HCF and LCM by prime factorization method, first we will find prime factors of given numbers.

Prime Factorization of 48:
48 = 2 × 24
= 2 × 2 × 12
= 2 × 2 × 2 × 6
= 2 × 2 × 2 × 2 × 3

Prime Factorization of 192:

192 = 2 × 96
= 2 × 2 × 48
= 2 × 2 × 2 × 24
= 2 × 2 × 2 × 2 × 12
= 2 × 2 × 2 × 2 × 2 × 6
= 2 × 2 × 2 × 2 × 2 × 2 × 3

HCF of 48 and 192 by prime factorization method:

Common factors in above prime factors of given numbers are underlined.

Common prime factors = 2, 2, 2, 2, 3

Now we have to multiply these common prime factors to obtain the HCF of given numbers.

HCF = 2 × 2 × 2 × 2 × 3
= 48

∴ HCF(48, 192) = 48

LCM of 48 and 192 by prime factorization method:

Now to find the LCM we will note down how many times the each factor has occurred in above prime factors of given numbers.

In above table we have also noted the maximum occurrence of the each factor in the prime factors of the given numbers.

Now to obtain the LCM of given numbers we will multiply each factor maximum number of times it occurred in above table.

LCM = 2 × 2 × 2 × 2 × 2 × 2 × 3
= 192

∴ LCM(48, 192) = 192

Division Method:

HCF of 48 and 192 by division method:

In the above division, the last divisor is 48.

Hence the HCF(GCF) of 48 and 192 = 48

∴ HCF(48, 192) = 48

LCM of 48 and 192 by division method

∴ LCM(48, 192) = 192

HCF of 48 and 192 by Listing Factors Method:

To find HCF by listing factors method we will list down all the factors of the given numbers.

Factors of 48:

12346812162448

Factors of 192:

12346812162432486496192

The greatest common factor in the above lists will be the HCF of the given numbers.

48 is the greatest factor which is common in the above lists.

∴ HCF(48, 192) = 48

LCM of 48 and 192 by Listing Multiples Method:

To find LCM by listing multiples method we will list down the multiples of given numbers.

Multiples of 48:

4896144192240288336384432480

Multiples of 192:

19238457676896011521344153617281920

The lowest common multiple in the above lists will be the LCM of the given numbers.

192 is the lowest multiple which is common in the above lists.

∴ LCM(48, 192) = 192

If the HCF of two numbers is 48 and their product is 9216, what is their LCM?Solution:If the LCM of two numbers is 192 and their product is 9216, what is their HCF?Solution:HCF and LCM of two numbers is 48 and 192 respectively. If one number is 192, what is the other number?Solution:

Let other number be x.
Given:
HCF(192, x) = 48
LCM(192, x) = 192
∵ HCF × LCM = product of numbers
∴ 48 × 192 = 192 × x
∴ 9216 = 192x
∴ 9216⁄192 = x
∴ x = 48
Hence, the other number is 48.

Solution: