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What is HCF and LCM?

HCF (Highest Common Factor)

HCF is the largest number that divides two or more positive integers without leaving a remainder.

It is also known as the greatest common divisor (GCD).

For example, the HCF of 4, 6 and 8 is 2 as 2 is the highest common factor that can divide 4, 6 and 8.

4 = 2 x 2 6 = 3 x 2 8 = 4 x 2

LCM (Least Common Multiple)

LCM of two or more numbers is the smallest number that is a common multiple of the given numbers.

It can be found by multiplying the numbers together and then dividing by the HCF.

For example, the LCM of 12 and 18 is 36, because 36 is the smallest number that is a multiple of both 12 and 18.

The multiples of 12 are 12, 24, 36, 48, 60, 72 and so on. The multiples of 18 are 18, 36, 54, 72 and so on.

We can see, that 36 is the smallest first common value among these multiples. Hence, 36 is the Least Common Multiple (LCM) for 12 and 18.

Properties of HCF and LCM

Property 1

The product of the LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers. This holds true for any pair of natural numbers.

LCM × HCF = Product of the numbers If A and B are two numbers, then LCM (A & B) × HCF (A & B) = A × B

Example: If 6 and 8 are two numbers.

Solution: LCM (6, 8) = 24 Steps to find LCM for 6 and 8.

Step 1: First few multiples of 6 and 8 are:

Multiples of 6: 6, 12, 18, 24, 30 Multiples of 8: 8, 16, 24, 32, 40

Step 2: LCM is the smallest number that appears in both lists: LCM (6, 8) = 24

HCF (6, 8) = 2 Steps to find HCF of 6 and 8

Factors of 6 are 1, 2, 3, and 6. Factors of 8 are 1, 2, 4, and 8. We can see the highest common factor is 2.

Therefore, LCM (6, 8) x HCF (6, 8) = 24 x 2 = 48 Also, 6 x 8 = 24 Hence, proved.

Property 2

HCF of co-prime numbers is 1. Therefore, the LCM of given co-prime numbers is equal to the product of the numbers. i.e., LCM of Co-prime Numbers = Product of the numbers

Example: Take two coprime numbers, 21 and 22.

LCM of 21 and 22 = 462 Write down multiples of each number until you find the first common multiple:

The multiples of 21 are: 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 462, 483, and so on

The multiples of 22 are: 22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 242, 264, 286, 308, 330, 352, 374, 396, 418, 440, 462, 484, and so on

From the above, we can say that LCM (21, 22) = 462

Product of 21 and 22 = 21 x 22 = 462

Therefore, according to the property, LCM (21, 22) = 21 x 22

Property 3

The highest common factor (HCF) of any given number will never be bigger than any of the numbers.

For example, we have two numbers 20 and 15. The factors of 20 include 20, 10, 5, 4, 2, and 1. And the factors of 15 include: 15, 5, 3, 1.

So, for 15 and 20, the HCF is 5 which is lesser than both the numbers 20 and 15.

Property 4

The lowest common multiple of any given number will never be lesser than any of the numbers.

We have two numbers 25 and 15. For 25 and 15, the LCM is 75.

So, the LCM of 25 and 15 is 75 which is greater than both the numbers 25 and 15.

Property 5

Finding HCF and LCM for the fraction values.

Consider the following two fractions: (a/b) and (c/d) The generalised formula for calculating the LCM and HCF of (a/b) and (c/d) is given below:

HCF of (a/b) and (c/d) = HCF (a, c)/ LCM (b, d) LCM of (a/b) and (c/d) = LCM (a, c)/ HCF (b, d) i.e., HCF = HCF of Numerators / LCM of Denominators LCM = LCM of Numerators / HCF of Denominators

For example, we have two fractions 4/7 and 12/ 5. The HCF of 4/7 and 12/5 = HCF (4, 12) / LCM (7, 5) = 4/35 The LCM of 4/7 and 12/5 = LCM (4, 12) / HCF (7, 5) = 12/1 = 12

Methods to find HCF and LCM

The common methods to solve HCF and LCM are:

Prime factorisation method

Division method

Prime Factorisation Method

HCF by Prime Factorisation

There are three important steps to be followed for this method:

Step 1: We need to find the prime factors of each of the given numbers. Step 2: We need to identify the common prime factors of the given numbers. Step 3: We then multiply the common factors obtained in step 2.

Therefore, the product of these common factors is the HCF of the given numbers.

For example, to find the HCF of 24 and 36, we can list the prime factors of each number:

24 = 2 x 2 x 2 x 3 = 2^{3} x 3^{1} 36 = 2 x 2 x 3 x 3 = 2^{2} x 3^{2}

The HCF is the highest power of each common prime factor. In this case, the HCF is 2^{2} x 3^{1} = 12.

LCM by Prime Factorisation

There are three important steps to be followed for this method:

Step 1: We need to find the prime factors of each of the given numbers. Step 2: We need to identify the maximum number of times each prime factor appears from all the factors. Step 3: The product of the prime factors occurring in maximum numbers is the LCM of the given numbers.

Step 2: The product of all the factors with the highest powers: 2^{4} × 7

Step 3: 2 × 2 × 2 × 2 ×7 = 112

Division Method

HCF by Division Method

Another method to find the HCF is to use the division method. Simple steps to be followed to use this method:

Step 1: If two numbers are given, take the smaller number as the divisor and the larger number as a dividend.

Step 2: Perform division using the numbers in step 1. If you get the remainder as 0, then the divisor is the HCF of the given numbers.

Step 3: Suppose you get a remainder other than 0, then take the remainder as the new divisor and the previous divisor as the new dividend.

Step 4: Perform steps 2 and step 3 until you get the remainder as 0.

Example: Find the HCF of 24 and 15.

Step 1: Take 24 as a dividend and 15 as a divisor.

Step 2: We can divide the larger number by the smaller number and then divide the remainder by the smaller number until a remainder of 0 is reached.

Step 3: The last divisor is the HCF, which is 3 in this case.

LCM by Division Method

Simple steps to be followed:

Step 1: Consider the given numbers, and check whether the given numbers are divisible by 2 or not.

Step 2: If the number is divisible by 2 then divide and again check for the same.

Step 3: If the numbers are not divisible by 2 then check 3, and so on.

Step 4: Perform step 2 until you get 1 in the end.

Example: Find the LCM of 36 and 48.

Using the above steps, performing LCM of 36 and 48 as shown below.

Hence LCM = 2 x 2 x 2 x 2 x 3 x 3 x 3 = 432

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