The hcf and lcm of two numbers are 9 and 504 if one of the number is 63 find the other number

LCM of 36 and 63 is the smallest number among all common multiples of 36 and 63. The first few multiples of 36 and 63 are (36, 72, 108, 144, . . . ) and (63, 126, 189, 252, 315, 378, . . . ) respectively. There are 3 commonly used methods to find LCM of 36 and 63 - by division method, by listing multiples, and by prime factorization.

What is the LCM of 36 and 63?

Answer: LCM of 36 and 63 is 252.

Explanation:

The LCM of two non-zero integers, x(36) and y(63), is the smallest positive integer m(252) that is divisible by both x(36) and y(63) without any remainder.

Methods to Find LCM of 36 and 63

The methods to find the LCM of 36 and 63 are explained below.

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

LCM of 36 and 63 by Division Method

To calculate the LCM of 36 and 63 by the division method, we will divide the numbers(36, 63) by their prime factors (preferably common). The product of these divisors gives the LCM of 36 and 63.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 36 and 63. Write this prime number(2) on the left of the given numbers(36 and 63), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (36, 63) 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 36 and 63 is the product of all prime numbers on the left, i.e. LCM(36, 63) by division method = 2 × 2 × 3 × 3 × 7 = 252.

LCM of 36 and 63 by Prime Factorization

Prime factorization of 36 and 63 is (2 × 2 × 3 × 3) = 22 × 32 and (3 × 3 × 7) = 32 × 71 respectively. LCM of 36 and 63 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 32 × 71 = 252.
Hence, the LCM of 36 and 63 by prime factorization is 252.

LCM of 36 and 63 by Listing Multiples

To calculate the LCM of 36 and 63 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 36 (36, 72, 108, 144, . . . ) and 63 (63, 126, 189, 252, 315, 378, . . . . )
• Step 2: The common multiples from the multiples of 36 and 63 are 252, 504, . . .
• Step 3: The smallest common multiple of 36 and 63 is 252.

∴ The least common multiple of 36 and 63 = 252.

☛ Also Check:

LCM of 36 and 63 Examples

1. Example 1: Verify the relationship between GCF and LCM of 36 and 63.

   Solution:

   The relation between GCF and LCM of 36 and 63 is given as, LCM(36, 63) × GCF(36, 63) = Product of 36, 63

   Prime factorization of 36 and 63 is given as, 36 = (2 × 2 × 3 × 3) = 22 × 32 and 63 = (3 × 3 × 7) = 32 × 71

   LCM(36, 63) = 252 GCF(36, 63) = 9 LHS = LCM(36, 63) × GCF(36, 63) = 252 × 9 = 2268 RHS = Product of 36, 63 = 36 × 63 = 2268 ⇒ LHS = RHS = 2268

   Hence, verified.

Show Solution >

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The LCM of 36 and 63 is 252. To find the least common multiple (LCM) of 36 and 63, we need to find the multiples of 36 and 63 (multiples of 36 = 36, 72, 108, 144 . . . . 252; multiples of 63 = 63, 126, 189, 252) and choose the smallest multiple that is exactly divisible by 36 and 63, i.e., 252.

Which of the following is the LCM of 36 and 63? 252, 25, 42, 15

The value of LCM of 36, 63 is the smallest common multiple of 36 and 63. The number satisfying the given condition is 252.

If the LCM of 63 and 36 is 252, Find its GCF.

LCM(63, 36) × GCF(63, 36) = 63 × 36 Since the LCM of 63 and 36 = 252 ⇒ 252 × GCF(63, 36) = 2268

Therefore, the greatest common factor (GCF) = 2268/252 = 9.

What are the Methods to Find LCM of 36 and 63?

The commonly used methods to find the LCM of 36 and 63 are:

• Division Method
• Listing Multiples
• Prime Factorization Method

How to Find the LCM of 36 and 63 by Prime Factorization?

To find the LCM of 36 and 63 using prime factorization, we will find the prime factors, (36 = 2 × 2 × 3 × 3) and (63 = 3 × 3 × 7). LCM of 36 and 63 is the product of prime factors raised to their respective highest exponent among the numbers 36 and 63.
⇒ LCM of 36, 63 = 22 × 32 × 71 = 252.

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What is the Least Common Multiple (LCM)?

In mathematics, the least common multiple, also known as the lowest common multiple of two (or more) integers a and b, is the smallest positive integer that is divisible by both. It is commonly denoted as LCM(a, b).

Brute Force Method

There are multiple ways to find a least common multiple. The most basic is simply using a "brute force" method that lists out each integer's multiples.

As can be seen, this method can be fairly tedious, and is far from ideal.

Prime Factorization Method

A more systematic way to find the LCM of some given integers is to use prime factorization. Prime factorization involves breaking down each of the numbers being compared into its product of prime numbers. The LCM is then determined by multiplying the highest power of each prime number together. Note that computing the LCM this way, while more efficient than using the "brute force" method, is still limited to smaller numbers. Refer to the example below for clarification on how to use prime factorization to determine the LCM:

Greatest Common Divisor Method

A third viable method for finding the LCM of some given integers is using the greatest common divisor. This is also frequently referred to as the greatest common factor (GCF), among other names. Refer to the link for details on how to determine the greatest common divisor. Given LCM(a, b), the procedure for finding the LCM using GCF is to divide the product of the numbers a and b by their GCF, i.e. (a × b)/GCF(a,b). When trying to determine the LCM of more than two numbers, for example LCM(a, b, c) find the LCM of a and b where the result will be q. Then find the LCM of c and q. The result will be the LCM of all three numbers. Using the previous example:

Note that it is not important which LCM is calculated first as long as all the numbers are used, and the method is followed accurately. Depending on the particular situation, each method has its own merits, and the user can decide which method to pursue at their own discretion.

We will learn the relationship between H.C.F. and L.C.M. of two numbers.

First we need to find the highest common factor (H.C.F.) of 15 and 18 which is 3.

Then we need to find the lowest common multiple (L.C.M.) of 15 and 18 which is 90.

H.C.F. × L.C.M. = 3 × 90 = 270

Also the product of numbers = 15 × 18 = 270

Therefore, product of H.C.F. and L.C.M. of 15 and 18 = product of 15 and 18.

Again, let us consider the two numbers 16 and 24

Prime factors of 16 and 24 are:

         16 = 2 × 2 × 2 × 2

         24 = 2 × 2 × 2 × 3

L.C.M. of 16 and 24 is 48;

H.C.F. of 16 and 24 is 8;

L.C.M. × H.C.F. = 48 × 8 = 384

Product of numbers = 16 × 24 = 384

So, from the above explanations we conclude that the product of highest common factor (H.C.F.) and lowest common multiple (L.C.M.) of two numbers is equal to the product of two numbers

or, H.C.F. × L.C.M. = First number × Second number

or, L.C.M. = \(\frac{\textrm{First Number} \times \textrm{Second Number}}{\textrm{H.C.F.}}\)

or, L.C.M. × H.C.F. = Product of two given numbers

or, L.C.M. = \(\frac{\textrm{Product of Two Given Numbers}}{\textrm{H.C.F.}}\)

or, H.C.F. = \(\frac{\textrm{Product of Two Given Numbers}}{\textrm{L.C.M.}}\)

Solved examples on the relationship between H.C.F. and L.C.M.:

1. Find the L.C.M. of 1683 and 1584.

Solution:

First we find highest common factor of 1683 and 1584                      

Therefore, highest common factor of 1683 and 1584 = 99

Lowest common multiple of 1683 and 1584 = First number × Second number/ H.C.F.

                                                               = \(\frac{1584 × 1683}{99}\)

                                                               = 26928

2. Highest common factor and lowest common multiple of two numbers are 18 and 1782 respectively. One number is 162, find the other.

Solution:

We know, H.C.F. × L.C.M. = First number × Second number then we get,

18 × 1782 = 162 × Second number

\(\frac{18 × 1782}{162}\) = Second number

Therefore, the second number = 198

3. The HCF of two numbers is 3 and their LCM is 54. If one of the numbers is 27, find the other number.

Solution:

HCF × LCM = Product of two numbers

3 × 54 = 27 × second number

Second number = \(\frac{3 × 54}{27}\)

Second number = 6

4. The highest common factor and the lowest common multiple of two numbers are 825 and 25 respectively. If one of the two numbers is 275, find the other number.

Solution:

We know, H.C.F. × L.C.M. = First number × Second number then we get,

                        825 × 25 = 275 × Second number

                \(\frac{825 × 25}{275}\) = Second number

Therefore, the second number = 75

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● Multiples.

Common Multiples.

Least Common Multiple (L.C.M).

To find Least Common Multiple by using Prime Factorization Method.

Examples to find Least Common Multiple by using Prime Factorization Method.

To Find Lowest Common Multiple by using Division Method

Examples to find Least Common Multiple of two numbers by using Division Method

Examples to find Least Common Multiple of three numbers by using Division Method

Relationship between H.C.F. and L.C.M.

Worksheet on H.C.F. and L.C.M.

Word problems on H.C.F. and L.C.M.

Worksheet on word problems on H.C.F. and L.C.M.

5th Grade Math Problems

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