If the H.C.F. and LCM of two numbers are 12 and 180 what may be the Lowest sum of those two numbers

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The highest common factor (HCF) of two or more given numbers is the largest number which divides each of the given numbers without leaving any remainder. The lowest common multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers. It is very important to learn HCF and LCM in mathematics as it helps us to do our day-to-day problems related to grouping and sharing. Let's learn about the different methods used to find the HCF and LCM of numbers.

What is HCF and LCM?

HCF is defined as the highest common factor present in two or more given numbers. It is also termed as the "Greatest Common Divisor" (GCD). For example, the HCF of 24 and 36 is 12, because 12 is the largest number which can divide both the numbers completely. Similarly, the least common multiple (LCM) of two or more numbers is the smallest number which is a common multiple of the given numbers. For example, let us take two numbers 8 and 16. Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and so on. The multiples of 16 are 16, 32, 48, 64, 80, 96, and so on. The first common value among these multiples is the least common multiple (LCM) for 8 and 16, which is 16. Now, let us learn two commonly used methods to find HCF and LCM.

How to Find HCF and LCM?

There are various methods to find the HCF and LCM of numbers. The most common methods are:

• Prime factorization method
• Division method

Let us discuss these methods in detail.

Finding HCF and LCM by Prime Factorization

By using the prime factorization method for finding LCM and HCF, we first need to find the prime factors of the given numbers by using either the ladder method or the factor tree method. Then, we can calculate the values of HCF and LCM by following the process explained below.

HCF by Prime Factorization

To find the HCF of the given numbers by prime factorization, we find the prime factors of those numbers. After finding the factors, we find the product of the prime factors that are common to each of the given numbers. For example, let us find the HCF of 50 and 75 by the prime factorization method.

• The prime factors of 50 = 2 × 5 × 5
• The prime factors of 75 = 3 × 5 × 5

The common factors of 50 and 75 are 5 × 5. Thus, HCF of (50, 75) = 25.

LCM by Prime Factorization

To calculate the LCM of any given numbers using the prime factorization method, we follow the steps given below:

• Step 1: List the prime factors of the given numbers and note the common prime factors.
• Step 2: The LCM of the given numbers = product of the common prime factors and the uncommon prime factors of the numbers.

Note: Common factors will be included only once.

Let us find the LCM of 160 and 90 using prime factorization.

• Step 1: The prime factors of 160 = 2 × 2 × 2 × 2 × 2 × 5 and 90 = 2 × 3 × 3 × 5.
• Step 2: The product of all the prime factors = Common prime factors (2 × 5) × Uncommon prime factors (2 × 2 × 2 × 2 × 3 × 3) = 1440.

Therefore, LCM of 160 and 90 = 1440.

Finding HCF and LCM by Division Method

There are two different ways to apply the division method to find LCM and HCF. Let us learn it one by one.

HCF by Division Method

To find the HCF by division method, follow the steps given below:

• Step 1: First, we need to divide the larger number by the smaller number and check the remainder.
• Step 2: Make the remainder of the above step as the divisor and the divisor of the above step as the dividend and perform the division again.
• Step 3: Continue the division process till the remainder is not equal to 0.
• Step 4: The last divisor will be the HCF of the given numbers.

Let's understand this method using an example. Here, we will find HCF of 198 and 360 using the division method. Read out the following steps and relate them with the image below.

• Divide 360 by 198. The obtained remainder is 162.
• Make 162 as the divisor and 198 as the dividend and perform the division again. Here the obtained remainder is 36.
• Make 36 as the divisor and 162 as the dividend and perform the division again. Here the obtained remainder is 18.
• Make 18 as the divisor and 36 as the dividend and perform the division again. Here the obtained remainder is 0.
• The last divisor, 18, is the HCF of 360 and 198.

To find the LCM of numbers by the division method, we divide the numbers with prime numbers and stop the division process when we get only 1 in the final row. Observe the steps given below to find the LCM of the given numbers using the division method.

• Step 1: Divide the numbers by the smallest prime number such that the prime number should at least divide 1 given number.
• Step 2: Write the quotients right below the numbers in the next row.
• Step 3: Now, for the next division step consider the above quotients as the new dividends.
• Step 4: Think of a prime number again which exactly divides at least 1 of the dividends.
• Step 5: Repeat the steps till we get 1 in the final row.
• Step 6: Multiply all the prime numbers on the left-hand side of the bar. That will be the required LCM.

Let us take an example of four numbers: 7, 8, 14, and 21, and follow the steps written below:

• Step 1: Divide the numbers 7, 8, 14, 21 by the smallest prime number, i.e., 2.
• Step 2: Write the quotients of the divisible numbers (8 and 14) below the numbers in the next row in this way: 7, 4, 7, and 21. Note, that for the remaining numbers 7 and 21, which are not divisible by 2, we copy the numbers as it is.
• Step 3: Now for the next division step, 7, 4, 7, 21 will be the new dividends.
• Step 4: Think of the smallest prime number again which exactly divides at least one of the numbers 7, 4, 7, 21.
• Step 5: Repeat the process and write the quotient below the numbers. Here, on dividing 7, 4, 7, 21 by 2, we get the quotients as 7, 2, 7, 21. [Only 4 was divisible by 2 in this step, so we copy the other three numbers as it is in the next row]
• Step 6: Now, 7, 2, 7, 21 are the next dividends.
• Step 7: Repeat the steps till we get 1.
• Step 8: Multiply all the prime numbers at the left-hand side of the bar to get the LCM of the given numbers.

Note: Divide the numbers only by prime numbers.

Do you know that for any two numbers, if we know any one of the values of HCF or LCM, we can easily find the other without using any of the above 2 methods? LCM and HCF of two numbers share a relationship with them and with each other that we are going to learn now.

HCF and LCM Formula

The LCM and HCF formula of two numbers 'a' and 'b' is given as HCF × LCM = a × b. In other words, the formula of HCF and LCM states that the product of any two numbers is equal to the product of their HCF and LCM. To know more about LCM and HCF relationship, visit this article.

HCF and LCM Tricks:

• If 1 is the HCF of 2 numbers, then their LCM will be their product.
• For two coprime numbers, the HCF is always 1.

Difference between HCF and LCM

The difference between the concept of HCF and LCM will be cleared to you through the following table:

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Let us have a look at some solved examples on HCF and LCM.

1. Example 1: Find the HCF and LCM of 180 and 24 using prime factorization.

   Solution:

   HCF of 180 and 24:
   The prime factors of 180 = 2 × 2 × 3 × 3 × 5 The prime factors of 24 = 2 × 2 × 2 × 3 The HCF is the product of the common prime factors of the given numbers.

   Therefore, the HCF of 180 and 24 is 2 × 2 × 3 = 12.

LCM of 180 and 24:

The prime factors of 180 = 2 × 2 × 3 × 3 × 5 The prime factors of 24 = 2 × 2 × 2 × 3

Taking all the prime factors from both the numbers only once, we have: Common prime factors (2 × 2 × 3) × Uncommon prime factors (2 × 3 × 5) = 360

Example 2. Find the HCF and LCM of 126 and 162 using the division method.

Solution: First, we will find the HCF of the two numbers 126 and 162 using the given steps:

Divide 162 by 126. The obtained remainder is 36.

Make 36 as the divisor and 126 as the dividend and perform the division again. Here the obtained remainder is 18.

Make 18 as the divisor and 36 as the dividend and perform the division again. Here the obtained remainder is 0.

The last divisor,18, is the HCF of 126 and 162.

Therefore, the HCF of 126 and 162 = 18.

Let us find the LCM of 126 and 162 by division method using the following steps:

• Step 1: Divide the numbers 126 and 162 by the smallest prime number, i.e., 2.
• Step 2: Write the quotient below the numbers in the next row: 63 and 81.
• Step 3: Now for the next division step, 63 and 81 will be the dividends.
• Step 4: Think of a prime number again which divides at least one of the two numbers 63 and 81.
• Step 5: Write the quotient below the numbers 63 and 81. The next set of quotients are 21 and 27.
• Step 6: Now 21 and 27 are the new dividends.
• Step 7: Repeat the steps till we get 1 in the final row.
• Step 8: Multiply all the prime numbers at the left-hand side of the bar and get the LCM of the given numbers.

Therefore, the LCM of 126 and 162 is 1134.

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FAQs on HCF and LCM

The full form of HCF is "Highest Common Factor" and the full form of LCM is "Least Common Multiple" or "Lowest Common Multiple".

What is the Difference Between HCF and LCM?

The least common multiple (LCM) of two or more numbers is the smallest number among all the common multiples of the given numbers, whereas, the HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers.

What is the Relationship Between HCF and LCM of Two Numbers?

The relationship between the HCF and LCM of two numbers is that the product of the LCM and HCF of any two given numbers is equal to the product of the given numbers. Let us assume 'a' and 'b' are the two given numbers. The formula that shows the relationship between their LCM and HCF is: LCM (a,b) × HCF (a,b) = a × b. For example, let us take two numbers 12 and 8. Let us use the formula: LCM (12,8) × HCF (12,8) = 12 × 8. The LCM of 12 and 8 is 24; and the HCF of 12 and 8 is 4. Putting the values in the formula we have 24 × 4 = 12 × 8. This shows: 96 = 96.

What is the HCF and LCM of numbers?

The highest common factor (HCF) of the given numbers is the largest number which divides each of the given numbers without leaving any remainder. The least common multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers.

What is the Use of HCF and LCM?

HCF can be used in the following situations:

• When we want to divide the things into smaller sections.
• To arrange things in groups and rows.

LCM can be used in the following situations:

• An event that is repeating continuously.
• For the analysis of a situation that will occur again at the same time.

How do you find the HCF and LCM in Math?

There are various methods to find the HCF and LCM of numbers. The two common ways to find the LCM and HCF of the given numbers are the prime factorization method and the division method.

What are the Steps to be Followed to Calculate the HCF and LCM of Two Numbers Using the Division Method?

To find the HCF of the given numbers by division method, we follow the given steps:

• Divide the given numbers (larger number by the smaller number) and check the remainder.
• Make the remainder of the above step as the divisor; and the divisor of the above step as the dividend and perform the division again.
• Continue the division process till we get the remainder as 0.
• The last divisor will be the HCF of the two numbers.

To find the LCM of the given numbers by division method we follow the given steps:

• Step 1: Divide the numbers by the smallest prime number.
• Step 2: Write the quotients right below the numbers in the next row.
• Step 3: Now, for the next division step consider the above quotients as the new dividends.
• Step 4: Think of a prime number again which exactly divides at least one of the dividends.
• Step 5: Repeat the steps till we get 1 in the final row.
• Step 6: Multiply all the prime numbers at the left hand side of the bar to get the LCM of the given numbers.

How to Find HCF and LCM using Prime Factorization?

Firstly, we find the prime factorization of the numbers. Then, the HCF of the given numbers will be the product of the common prime factors that occur in the prime factorization of both the numbers. And the LCM of those numbers will be the product of the common factors (taken only once) and the uncommon or the remaining factors.