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NCERT Solutions for Class 10 Hindi Sanchayan संचयन भाग 2

NCERT Solutions for Class 10 Hindi Sanchayan संचयन भाग 2

NCERT Solutions for Class 10 Hindi Sanchayan संचयन भाग 2

• Chapter 1 हरिहर काका
• Chapter 2 सपनों के-से दिन
• Chapter 3 टोपी शुक्ला

Greatest Common Factor (GCF) Calculator is the most excellent handy calculator for calculating GCF!

Factors are the numbers that we can multiply to each other to get one different number like 4 x 5 = 20, here 4 and 5 are factors

It is also possible to have various factors of a number like

4 x 5 = 20, 2 x 10 = 20, 1 x 20 = 20

Factors of 20: 1, 2, 4, 5, 10, and 20

So when we are finding the factors of two numbers like

Factors of 12 of 1, 2, 3, 4, 6, and 12

Factors of 20 of 1, 2, 4, 5, 10, and 20

So here common factors are 1, 2, and 4

A common factor is a factor of two or more numbers.

Greatest Common Factor is just the biggest of the common factors. We can define it as:

The GCF is the greatest positive whole number from the set of a number that divides equally into all numbers with zero remainders.

So here if we are taking the previous example, then the Greatest Common Factor of 12 and 20 is 4.

Then, What is the Greatest Common Factor of 0?

We know that when we multiply any whole number to zero, it becomes zero so it is clear that each non zero whole number is a factor of 0. Like

n × 0 = 0 so, 0 ÷ n = 0, where n is any whole number

So GCF(n,0) = n, where n is any whole number

But, GCF(0, 0) is undefined.

There are so many ways available to find out the greatest common factor of any whole numbers such as Factoring, Prime Factorization, Euclid’s Algorithm, and many more. Which one of the methods is useful for you is decided by some factors like

1. How many numbers are there?
2. How long are the numbers?
3. What is the purpose of finding the GCF?

So, go for any of the methods and get your GCF. Let’s go through each method in detail.

Here, to check the Greatest Common Factor using the factoring method, we have to find out all the factors of each number manually or you can use any online Factors calculators also. So check out all the positive whole number factors of a number that can divide equally into the integers with zero remainders. Now write down all the common factors for each number, then find the biggest common number, it the GCF of numbers.

For detailed information see our Factoring Calculator.

Let’s check this with some examples to make it more simple,
Example 1:

Factors of 16: 1, 2, 4, 8, and 16

Factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24

The common factor of 16 and 24: 1, 2, 4, and 8

The greatest common factor of 16 and 24: 8.

Example 2:

Factors of 15: 1, 3, 5, and 15

Factors of 30: 1, 2, 3, 5, 6, 10, 15 and 30

Factors of 105: 1, 3, 5, 7, 15, 21, 35 and 105

Common factors of 15, 30 and 105: 1, 3, 5 and 15

Greatest Common Factor of 15, 30, and 105: 15.

Prime Factorization is helpful when numbers are larger. If you want to calculate the Greatest Common Factor using prime factorization, then you have to check out all of the prime factors of each number manually or you can check it with an online Prime Factors Calculator. Now from the prime factors, list out all the common prime factors of the numbers. Find out the maximum same numbers from each prime factor and multiply them to find out the Greatest Common Factor.

For detailed information see our Prime Factorisation Calculator.

Example 1: Find out GCF of 16 and 24?

Prime Factorization of 16 = 2 x 2 x 2 x 2

Prime Factorization of 24 = 2 x 2 x 2 x 3

The occurrence of common prime factor of 16 and 24: 2, 2 and 2

The greatest common factor of 16 and 24 is 2 x 2 x 2 = 8.

Example 2: Find out GCF of 15, 30 and 105

Prime factorization of 15 = 3 x 5

Prime factorization of 30 = 2 x 3 x 5

Prime factorization of 105 = 3 x 5 x 7

Common factors of 15, 30 and 105 = 3 and 5

Greatest Common Factor of 15, 30, and 105 = 3 x 5 = 15

Now, what if we are required to check the greatest common factor of very large numbers like 134334, 124456, or 187644? Some online calculators like Greatest Common Factor (GCF) or Factoring Calculator can help to find out the GCF of such a large number but what if you need to do it manually by yourself.

Step by step procedure to find the GCF of larger numbers with the help of Euclid’s Algorithm

1. From the given numbers, first of all, take 2 whole numbers, and from the big number subtract the smaller number and write the result.
2. Now subtract the small number from the result. Do it again and again till the result is less than the original small number.
3. Now use the small number as a larger number and subtract the result you got from step two. Repeat the same procedure for every new bigger number and smaller number till you get the 0.
4. Now if you get 0, then go back to one step before you get 0, the greatest common factor is the number you got just before the 0.

For detailed information see our Euclid’s Algorithm Calculator.

Let’s check it out with some examples.

Example 1: Find out GCF of 16 and 24

24 – 16 = 8

16 – 8 – 8 = 0

So, the GCF of 16 and 24 is 8, the least result we had before we got zero.

Example 2: Find out GCF of 15, 25 and 105

Here we have 3 numbers and for this, the method of finding GCF is:

GCF (x,y,z) = GCF (GCF (x,y),z)

So here we have to, first of all, find the greatest finding factor of 2 numbers and then we use its result with the 3rd number and find the GCF.

So Let’s get GCF (105, 25) first here,

105 – 25 = 80

80 – 25 = 55

55 – 25 = 30

30 – 25 = 5

25 – 5 = 20

20 – 5 – 5 – 5- 5 = 0

So, the greatest common factor of 105 and 25 is 5.

Now let’s check GCF of 3rd number, 15, and our result is also 5, GCF (5, 15)

15 – 5 – 5 – 5 = 0

So, the GCF of 15 and 5 is 5.

Therefore, the greatest common factor of 105, 25, and 15 is 5.

Example 3: Find the GCF 268442, 178296, and 66888

First of all, let’s find the GCF (268442, 178296)

268442 – 178296 = 90146

178296 – 90146 = 88150

90146 – 88150 = 1996

88150 – (1996 x 44) = 326

1996 – (326 x 6) = 40

326 – (40 x 8) = 6

40 – (6 x 6) = 4

6 – 4 = 2

4 – (2 x 2) =

So, the greatest common factor of 268442 and 178296 is 2.

Now we check the GCF (2, 66888)

66888 – (2 x 33444) = 0

So, the GCF of 2 and 66888: 2.

Hence, the GCF of 1268442, 178296, and 66888 is 2

Prime Factorization Calculator used to calculate the prime factorization of the number you enter here. (Numbers above 10 million may or may not time out. When you have to find out the Prime Factorization of very large numbers, the Prime Factorization Calculator can be very handy and useful.

The prime factorization calculator can help you to find prime factors of any integer number up to 1 trillion. Just enter the number here in the calculator and within no time you’ll get the prime factorization.

Prime Factorization Calculator various applications:

• Check Prime factors of an integer number up to 1 trillion
• Also, it can check either a number is prime or not
• Prime decomposition in an exponential way
• Make a list of all positive prime numbers smaller than the number you entered and calculate them
• Comma-separated Values (CSV) of prime factors

To know the complete process, first of all, you must know all about what is a prime number?

A prime number is a positive whole number that is greater than 1 and that can be divided equally just by one and by itself. We can also say that cannot be achieved by multiplying 2 smaller numbers than that. Like 7 is a prime number! You can divide it by 1 or 7 only.

if an integer number is small then it’s very simple to decide whether a number is prime or not. For small numbers, we can use the rules of divisibility and trial division manually. But what if a number is very large and we have to determine whether it is prime or not. It is hard to check it manually as factoring large numbers carry very large primes. In such situations, an online calculator is the best to choose.

What is the prime factor?

Factors of prime numbers are prime factors. So here if we consider the factors of 12, that is, we need to get what integer numbers multiply to provide you 12. We understand that 1 * 12 = 12, 2 * 6 = 12 and 3 * 4 = 12. But here 12, 6, and 4 are not prime factors. The only prime factors of 12 are 2 and 3. You can also check prime factors using a factor calculator.

What is Prime Factorization?

Any whole number can be uniquely described as the list of prime numbers and such representation is named the prime factorization, or canonical representation or integer factorization or prime decomposition. There is exactly one prime factorization for any positive integer number, and such positive integer has just 1 prime factorization.

How to Find Prime Factorization of a Number?

There are various processes to find out the Prime Factorization of a Number and you can choose any of one as per the need. Here, we try to include two of the methods of prime factorization:

1. Trial division and
2. Using prime factors trees.

Trial Division

One of the methods to check the Prime Factor of a number is trial division. Trial division consists of very easy and basic algorithms, though it is an extremely slow process. In this method, we have to check each number by dividing the composite number in question by the integer and deciding if, and how many times, the number can divide the number equally.

To make it more understandable, let go through an example:

To get the prime factorization of 48, we have to start with dividing it by the smallest prime 2

48 ÷ 2 = 24

24 ÷ 2 = 12

12 ÷ 2 = 6

6 ÷ 2 = 3

But, we can’t divide 3 by 2, but we can divide 3 by itself only

3 ÷ 3 = 1

So here he prime factorization of 48 = 2 X 2 X 2 X 2 X 3 = 2⁴  X 3

We can check it in a prime factorization calculator also. The algorithm used in the calculator and trial division may differ but the result is always the same.

Prime Decomposition

Another popular method to find prime factorization is known as prime decomposition and it includes the use of a factor tree. The factor tree diagram is an easy process to divide a number into its prime factors. To create a factor tree we have to break down the composite number into factors of the composite number till the numbers are prime.

There might be various methods to show the factor tree for any provided prime factorization.

Here let’s take an example of 48 to check the prime factor using prime decomposition method.

It doesn’t matter whether you are using a calculator or find yourself, the factorization will be unique always. And when there are very big numbers, the calculator is much more amazing!

FAQs on Prime Factorization

Question 1: What is Prime Factorization?

Answer 1: Prime factorization means getting all the prime numbers that are factors of a number.

Question 2: What is The Sieve of Eratosthenes?

Answer 2: Eratosthenes invented a sieve to find prime numbers. It generally removes composite numbers and keeps prime numbers.

Question 3: What is a Composite Number?

Answer 3: When an integer number has more than two factors it is known as a composite number.

Question 4: Why find Prime Factors?

Answer 4: Prime Numbers can not be broken down so they are the basic blocks of all numbers. and this can be very helpful when working with large numbers, like in Cryptography.

 

Chapter Wise NCERT Solutions for Class 10 Hindi Kshitij क्षितिज भाग 2

NCERT Solutions for Class 10 Hindi Kshitij (क्षितिज) भाग 2

NCERT Solutions for Class 10 Hindi Kshitij क्षितिज भाग 2

काव्य – खंड

• Chapter 1 पद
• Chapter 2 राम-लक्ष्मण-परशुराम संवाद
• Chapter 3 सवैया और कवित्त
• Chapter 4 आत्मकथ्य
• Chapter 5 उत्साह और अट नहीं रही
• Chapter 6 यह दंतुरहित मुस्कान और फसल
• Chapter 7 छाया मत छूना
• Chapter 8 कन्यादान
• Chapter 9 संगतकार

गद्य – खंड

• Chapter 10 नेताजी का चश्मा
• Chapter 11 बालगोबिन भगत
• Chapter 12 लखनवी अंदाज़
• Chapter 13 मानवीय करुणा की दिव्या चमक
• Chapter 14 एक कहानी यह भी
• Chapter 15 स्त्री शिक्षा के विरोधी कुतर्कों का खंडन
• Chapter 16 नौबतखाने में इबादत
• Chapter 17 संस्कृति

LCM of Two or More Numbers Calculator: Finding LCM of two or more numbers is a little tough for big numbers to make it simple just use our free LCM calculator. Give your inputs and press on the ‘Calculate LCM’ button to get the Least common multiple of certain numbers.

This Online Least Common Multiple LCM of Numbers Calculator is pretty simple to use and handy for all students to find the LCM of N numbers easily.

What is the LCM of Numbers?

Least Common Multiple is the smallest number of all common multiples. We can also determine it as:

The LCM of two or more integers like a,b,c is the least number that is evenly divisible by all numbers in the set.

For example, here we are taking the three integers i.e., a=10, b=15, c=20, the least common multiple of 10, 15, and 20 is 60.

Free & Handy Least Common Multiple Calculator for N numbers

LCM of Numbers is nothing but finding the least common multiple for a set of numbers like two, or more. If your inputs are more than two then this LCM Calculator will help you to calculate the Least multiple of those numbers within seconds and give you results with a detailed step by step explanation for your knowledge. Our handy LCM of numbers calculator is quite simple and user-friendly to use so any age of the students can quickly understand how it works.

All you need to do to find LCM for two or more numbers is just place the inputs by separating with commas like 100, 5000, 7500 and then click on Calculate LCM button. That’s it! The solution will be displayed on the screen with a neat explanation. Keep in mind that not to use any spaces or commas within numbers like 1000 4000 700 & 5,400, 10,500, 25,000.

Let’s dive into it and know the methods and steps to find LCM of two or more numbers

How to Find LCM of two or more numbers?

Well, we all know that there are different ways to calculate the LCM of numbers. Based on the numbers and their difficulty, we choose the method that suits us. So, practice all the methods by learning the procedure from here and find LCM of a set of numbers easily at any time.

Our LCM calculator mostly follow the three commonly used methods to find Least common multiple of two or more numbers and they are as under,

• Common Division Method
• GCF Formula Method
• Listing Multiples Method

Okay, let’s start knowing each of this method deeply with enough examples and understand the whole process of finding Least Common Multiple of a set of integers.

1. Finding LCM of two or more numbers using Common Division Method

In order to find out the LCM of Numbers by the Common Division Method, you should follow the below steps properly without any fail. So, let’s begin finding the Least Common Multiple with Common Division:

1. Place the inputs in a horizontal line, separating them by spaces or commas.
2. Divide the numbers with a prime number that fits and precisely divides at least two of the given integers.
3. Write the quotient below the numbers in the next row. If any of the numbers is not yet divided with the prime number, then directly bring it down in the next row.
4. Again divide the numbers with other prime numbers and bring the quotients in the next row. Continue this process till all co-prime integers left in the last row.
5. Now, multiple all the prime numbers that you have divided the inputs along with the co-prime numbers left in the last row. The product of all prime numbers is the LCM of given two or more numbers.

Example:

Find the LCM of numbers 10, 25, and 30?

Solution:

Given integers are 10, 25, and 30

Now, place these numbers in a row separated with commas and divide them with prime number i.e., 2.

2	10, 25, 30

Bring down the quotients in the next row and divide with another prime number that exactly divides the numbers.

2	10, 25, 30
5	5, 25, 15

Remain the same process until all co-prime numbers are left in the last row.

2	10, 25, 30
5	5, 25, 15
	1, 5, 3

Finally, multiply all the prime numbers by which we have divided and the co-prime numbers left in the last row i.e. 2 × 5 × 1 × 5 × 3 = 150.

Therefore, LCM of 10, 25, 30 is 150.

2. Finding the Least Common Multiple of a, b, c Using the GCF Formula

To calculate the LCM of two or more numbers like a, b, c by the greatest common factor (GCF) apply the formula which is mentioned below, and find the LCD of two or more numbers.

The formula of LCM by GCF is LCM(a₁,a₂,a₃….,a_n) = ( a₁ × a₂ × a₃ × …. × a_n) / GCF(a₁,a₂,a₃….,a_n).

First, you should find the GCF of the given numbers and then apply the answer to the LCM equation. Ay last, you will get the Least Common Divisor of two or more integers using the greatest common factor GCF.

Example: 

Find the Least common divisor or multiple of 10, 25, and 30 by GCF Formula

Solution:

The given numbers are 10, 25, 30

The formula is LCM

Place the given integers in the formula to find the LCM of 10, 25, 30

LCM(10, 25, 30)= ( 10 × 25 × 30 ) / GCF(10, 25, 30)

Now calculate the Greatest common factor (GCF) of 10, 25, and 30: GCF(10, 25, 30) = 5

Then, Apply 5 in the LCM equation to get the final result of LCD(10, 25, 30)

LCM(10, 25, 30)= ( 10 × 25 × 30 ) / GCF(10, 25, 30)
LCM(10, 25, 30) = ( 10 × 25 × 30 ) / 5
LCM(10, 25, 30) = 7500 / 5
LCM(10, 25, 30) = 150

Thus, the Least Common Divisor (LCD) or LCM of 10, 25, and 30 is 150.

3. Finding the Least Common Divisor (LCD) of  two or more numbers by Listing Multiples Method

To find the LCM of numbers more than two by listing multiples, initially, you should know what is a common multiple. The Common Multiples are those numbers that are found in all the lists of each number. Now, list the multiples for each given number till you found one of the common smallest multiples in all the lists. Thus, the smallest positive integer appears on all lists is the LCM of those numbers.

Example: 

Find the Lowest Common Multiple (LCM) of 10, 25, and 30

Solution:

First, List the multiples of 10, 25, 30 till one of the multiples appears on all lists.

Multiples of 10 is 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150

Multiples of 25 is 25, 50, 75, 100, 125, 150

Multiples of 30 is 30, 60, 90, 120, 150

Now, find the smallest positive integer that is appeared on all of the lists. For the sake of knowledge, we have bolded the common number in the list above.

Therefore, Least Common Multiple of 10, 25, 30 is 150.

Other Techniques to Find the Least Common Multiple (LCM) or (LCD)

1. Finding LCM of two or more numbers by Prime Factorization

Initially, calculate all the prime numbers of each given number. Now, List the Prime numbers that are common to each of the numbers. Multiply the list of prime factors together to find the LCM.

Example:

Find LCM of 10, 15, and 20 using prime factorization

Solution:

The given numbers are 10, 15, 20

The factors of each number are

10= 2 x 5

15= 3 x 5

20= 2 x 2 x 5

Multiply all factors except the repleted factors, to find the LCM. The LCM of 10, 15, 20 is 2 x 2 x 3 x 5 = 60.

Therefore LCM(10, 15, 2o) = 60.

2. How to Find LCD of numbers Using the Ladder Method (Cake Method)

This Ladder method uses division to calculate the Least Common Multiple of two or more numbers. Cake and Ladder Method is the easiest and quickest method to calculate the LCD of a set of numbers. Most of the people use this ladder method to find LCM of n numbers because it is a simple division.

Steps to Find the LCM by Cake or Ladder Method

1. Place your numbers in a cake layer (like a row).
2. Divide the layer numbers by a prime number that is exactly divisible into two or more numbers in the layer and take down the result into the next layer.
3. In case any number in the layer is not evenly divisible simply write it down directly in the next layer.
4. Repeat the step till dividing cake layers by prime numbers.
5. When there are no more primes that evenly divided into two or more numbers then you’re done with division calculation.
6. Now, calculate the product of the numbers in the L shape, left column, and bottom row. 1 is ignored. Then, the result number is LCM of two or more given numbers.

LCM Properties

• The LCM is associative: LCM(a, b) = LCM(b, a)
• The LCM is commutative: LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c))
• The LCM is distributive: LCM(da, db, dc) = dLCM(a, b, c)
• The LCM is related to the greatest common factor (GCF): LCM(a,b) = a × b / GCF(a,b) and GCF(a,b) = a × b / LCM(a,b)

FAQs on Least Common Division LCD or LCM Calculator of two or more numbers

1. How do you find LCD of two or more numbers?

You can find the Lowest Common Divisor or Multiple of given numbers by using various techniques. Commonly used methods of LCM are explained in a step by step manner above. Learn one particular method from all the methods and find an LCD of two or more numbers easily or else use our LCM of two or more numbers Calculator.

2. Where can I get solved examples for LCM of two or more numbers?

You can avail solved examples for all methods on our page to find the Least common multiple of two or more integers.

3. How do you find the LCM of two numbers on a calculator?

You can do it easily by using our Least common divisor calculator. Just give your inputs in the free LCM calculator and click on the ‘Calculate LCM’ button to find LCM of given numbers with detailed solution steps.

HCF Using Euclid’s Division Lemma Method: Finding the Highest Common Factor by Euclid’s Division Lemma Algorithm is a standard approach by all the students. Here, we will see the detailed process on How to Find HCF of two or more numbers by Euclid’s Division Lemma Algorithm.

What is Euclid’s Division Lemma?

The basis of the Euclidean division algorithm is Euclid’s division lemma. Euclid’s division algorithm is a method to calculate the Highest Common Factor (HCF) of two or three given positive numbers. Euclid’s Division Lemma says that for any two positive integers suppose a and b there exist two novel whole numbers say q and r, such that, a = bq+r, where 0≤r<b.

Here, a and b are given numbers whereas q and r are Quotient and Reminder.

To find the Highest Common Factor (HCF) of two positive integers a and b we use Euclid’s division algorithm. Highest Common Factor (HCF) of two or more numbers is the greatest common factor of the given set of numbers. If we consider two numbers to find the HCF using Euclid’s Division Lemma Algorithm then we need to choose the largest integer first to satisfy the statement, a = bq+r where 0 ≤ r ≤ b.

Let’s get deep to see how the algorithm works when finding the HCF of two or more given numbers.

How to Find the Highest Common Factor of given numbers Using Euclid’s Division Lemma?

Follow the below steps to find the HCF of given numbers with Euclid’s Division Lemma:

Step 1: Apply Euclid’s division lemma, to a and b. So, we find whole numbers, q and r such that a = bq + r, 0 ≤ r < b.
Step 2: If r = 0, b is the HCF of a and b. If r ≠ 0, apply the division lemma to d and r.
Step 3: Continue the process until the remainder is zero. The divisor at this stage will be the required HCF of a and b.

Thus, Euclid’s Division Lemma algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d,

Example: Use Euclid’s algorithm to find the HCF of 36 and 96.

Solution:

Given HCF of two numbers ie., 36 and 96.  The larger number from both a and b is 96, hence, apply the Euclid Division Lemma algorithm equation a = bq + r where 0 ≤ r ≤ b.

We have a= 96 and b= 36

⇒ 96 = 36 × 2 + 24, where 24≠0.

So, again apply the Euclid’s Division Algorithm for new dividend as 36 and divisor as 24

⇒ 36 = 24×1 +12, where 12≠0

Again take dividend as 24 and divisor as 12.

⇒ 24 = 12×2 +0, here reminder=0

As the remainder becomes zero, we can halt the process here itself. As per the Euclid’s division Lemma algorithm, the last divisor is 12.

Thus, the HCF of 36 and 96 is 12.

FAQs on HCF of two or more numbers by Euclid’s Division Lemma

1. What is meant by Euclid’s Division Lemma?

The definition of Euclid’s Division Lemma is if two positive integers say “a” and “b”, then there exists unique integers state “q” and “r” such that which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

2. What is Lemma?

Lemma is a proven statement used for proving another statement.

3. What represents Q and R in Euclid’s Division Lemma Technique?

The number ‘q’ is called the quotient and ‘r’ is called the remainder.

4. What is HCF of two or more numbers?

The full form of HCF is Highest Common Factor. The HCF of two or more given integers is defined as the greatest number which evenly divides a given set of numbers.

5. How to Find HCF using Euclid’s Division Lemma?

You can easily find HCF of a set of integers by Euclid’s division lemma along with a detailed explanation from our page. Enter the inputs and get the HCF of two or more numbers which is solved by using Euclid’s division lemma method with neat & understandable steps.