1. Rewrite \[$ 0.26^{\circ} \$\] As A Proper Fraction In The Form Of \[$\frac{a}{b}\$\].2. Questions: 2.1. Rewrite 92 And 146 As Products Of Their Prime Factors. 2.2. Write Down The LCM And HCF Of 92 And 146. 2.3. Divide 240

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1. Rewrite { 0.26^{\circ} $}$ as a proper fraction in the form of {\frac{a}{b}$}$.

To rewrite the given expression as a proper fraction, we need to convert the decimal to a fraction. We can do this by expressing the decimal as a ratio of two integers.

0.26 can be written as 26/100. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

26 ÷ 2 = 13 100 ÷ 2 = 50

So, 0.26 can be written as 13/50.

Therefore, { 0.26^{\circ} $}$ can be rewritten as {\frac{13}{50}$}$.

2. Questions

2.1. Rewrite 92 and 146 as products of their prime factors.

To rewrite 92 and 146 as products of their prime factors, we need to find the prime factors of each number.

Prime Factorization of 92

92 can be divided by 2, which is a prime number.

92 ÷ 2 = 46 46 ÷ 2 = 23

Since 23 is a prime number, we cannot divide it further.

Therefore, the prime factorization of 92 is 2^2 × 23.

Prime Factorization of 146

146 can be divided by 2, which is a prime number.

146 ÷ 2 = 73

Since 73 is a prime number, we cannot divide it further.

Therefore, the prime factorization of 146 is 2 × 73.

2.2. Write down the LCM and HCF of 92 and 146.

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of 92 and 146, we need to use their prime factorizations.

LCM of 92 and 146

The LCM of two numbers is the product of the highest powers of all the prime factors involved.

From the prime factorizations, we have:

92 = 2^2 × 23 146 = 2 × 73

The LCM will have the highest power of each prime factor.

LCM = 2^2 × 23 × 73

LCM = 4 × 23 × 73 LCM = 3388

HCF of 92 and 146

The HCF of two numbers is the product of the lowest powers of all the common prime factors.

From the prime factorizations, we have:

92 = 2^2 × 23 146 = 2 × 73

The HCF will have the lowest power of each common prime factor.

HCF = 2

2.3. Divide 240 by the HCF of 92 and 146.

To divide 240 by the HCF of 92 and 146, we need to find the HCF first.

The HCF of 92 and 146 is 2.

Now, we can divide 240 by 2.

240 ÷ 2 = 120

Therefore, 240 divided by the HCF of 92 and 146 is 120.

Discussion

In this article, we have rewritten the expression 0.26 as a proper fraction, found the prime factorizations of 92 and 146, and calculated their LCM and HCF. We have also divided 240 by the HCF of 92 and 146.

The prime factorization of a number is a way of expressing it as a product of prime numbers. It is a fundamental concept in number theory and has many applications in mathematics and computer science.

The LCM and HCF of two numbers are used in many mathematical operations, such as finding the greatest common divisor and the least common multiple.

In conclusion, this article has demonstrated the importance of prime factorization and the LCM and HCF in mathematics. It has also shown how to apply these concepts to solve problems and perform calculations.

References

Q1: What is prime factorization?

A1: Prime factorization is the process of expressing a number as a product of prime numbers. It is a way of breaking down a number into its simplest building blocks.

Q2: How do I find the prime factorization of a number?

A2: To find the prime factorization of a number, you can use the following steps:

  1. Start by dividing the number by the smallest prime number, which is 2.
  2. If the number is divisible by 2, continue dividing by 2 until it is no longer divisible.
  3. Then, move on to the next prime number, which is 3, and repeat the process.
  4. Continue this process until you have divided the number by all prime numbers up to its square root.
  5. If the number is still not fully divided, it is a prime number itself.

Q3: What is the difference between LCM and HCF?

A3: The LCM (Least Common Multiple) and HCF (Highest Common Factor) are two related but distinct concepts in mathematics.

  • The LCM of two numbers is the smallest number that is a multiple of both numbers.
  • The HCF of two numbers is the largest number that divides both numbers without leaving a remainder.

Q4: How do I find the LCM of two numbers?

A4: To find the LCM of two numbers, you can use the following steps:

  1. Find the prime factorization of each number.
  2. Identify the highest power of each prime factor that appears in either number.
  3. Multiply these highest powers together to get the LCM.

Q5: How do I find the HCF of two numbers?

A5: To find the HCF of two numbers, you can use the following steps:

  1. Find the prime factorization of each number.
  2. Identify the lowest power of each common prime factor that appears in both numbers.
  3. Multiply these lowest powers together to get the HCF.

Q6: What is the relationship between LCM and HCF?

A6: The LCM and HCF of two numbers are related by the following formula:

LCM(a, b) × HCF(a, b) = a × b

This formula shows that the product of the LCM and HCF of two numbers is equal to the product of the two numbers themselves.

Q7: How do I use prime factorization, LCM, and HCF in real-life situations?

A7: Prime factorization, LCM, and HCF have many practical applications in mathematics and computer science. Here are a few examples:

  • In cryptography, prime factorization is used to create secure encryption algorithms.
  • In computer science, LCM and HCF are used in algorithms for finding the greatest common divisor and the least common multiple.
  • In mathematics, prime factorization is used to solve problems in number theory and algebra.

Q8: Can you give me some examples of prime factorization, LCM, and HCF?

A8: Here are a few examples:

  • Prime factorization: 12 = 2^2 × 3
  • LCM: LCM(12, 15) = 60
  • HCF: HCF(12, 15) = 3

Q9: How do I practice prime factorization, LCM, and HCF?

A9: Here are a few tips for practicing prime factorization, LCM, and HCF:

  • Start with simple numbers and work your way up to more complex ones.
  • Use online resources and practice problems to help you learn.
  • Try to apply these concepts to real-life situations and problems.

Q10: Where can I learn more about prime factorization, LCM, and HCF?

A10: There are many online resources and textbooks that can help you learn more about prime factorization, LCM, and HCF. Some popular resources include:

  • Khan Academy: Prime Factorization
  • Math Is Fun: Prime Factorization
  • Wolfram Alpha: Prime Factorization

I hope this Q&A article has helped you understand prime factorization, LCM, and HCF better. If you have any more questions, feel free to ask!