Finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two numbers using prime factorization.
HCF using Prime Factorization:
The HCF, or Highest Common Factor, of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In other words, it is the greatest common divisor that the numbers share. The HCF is useful in various mathematical problems, including simplifying fractions, finding equivalent fractions, and solving certain types of equations.
Finding HCF:
Express each number as a product of its prime factors. This means breaking down each number into a multiplication of its prime factors. For example, if you want to find the HCF of 24 and 36, you would express them as
24 = 2^3 * 3^1 and
36 = 2^2 * 3^2.
Identify the common prime factors and their lowest powers in both numbers. Look at the prime factorization of each number and identify which prime factors are common to both numbers. In our example, both 24 and 36 have 2 and 3 as prime factors. Then, pick the lowest power of each common prime factor. In this case, 2 is squared in 36 and 3 is raised to the power of 1 in 24.
Multiply these common prime factors together to find the HCF. After identifying the common prime factors and their lowest powers, multiply these together to find the HCF. In our example, the common factors are 2^2 and 3^1, so the HCF is 2^2 * 3^1 = 12.
LCM using Prime Factorization:
The LCM, or Lowest Common Multiple, of two or more numbers is the smallest multiple that is exactly divisible by each of the numbers. In other words, it is the smallest positive integer that is a multiple of all the numbers under consideration. The LCM is used in various mathematical operations, such as adding and subtracting fractions with different denominators, as well as in solving problems related to periodic events and cycles.
Finding LCM:
Express each number as a product of its prime factors. Similar to finding the HCF, express each number as a product of its prime factors.
Identify all the prime factors and their highest powers in both numbers. Look at the prime factorization of each number and identify all the prime factors in both numbers and pick the highest power of each prime factor.
- LCM: All factors are 2^3, 3^2, so LCM = 2^3 * 3^2 = 72
Multiply these prime factors together to find the LCM. After identifying all the prime factors and their highest powers, multiply these together to find the LCM.
By following these steps, you can find the HCF and LCM of two numbers using prime factorization. This method is efficient and provides a systematic approach to finding these important concepts in number theory.
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