## LCM (LOWEST COMMON MULTIPLE)

In mathematics, **the LCM (Least Common Multiple) of two or more integers is the smallest positive integer that is divisible by all of them without leaving a remainder**. For example, the **LCM of 4 and 6 is 12** because 12 is the smallest positive integer that is divisible by both 4 and 6 without leaving a remainder.

Finding the LCM of two or more integers can be done in several ways, but the most common method is to use the prime factorization of each integer. To find the LCM of two or more integers using prime factorization, we take the highest power of each prime factor that appears in any of the prime factorizations and multiply them together.

For example, to find the **LCM of 12 and 20**, we first find their prime factorizations: 12 = 2^2 \times 3 and 20 = 2^2 \times 5. Then, we take the highest power of each prime factor that appears in either of the factorizations and multiply them together:

The highest power of 2 that appears is 2^2.

The highest power of 3 that appears is 3^1.

The highest power of 5 that appears is 5^1.

Therefore, the LCM of 12 and 20 is 2^2 \times 3^1 \times 5^1 = 60.

**See also: FACTORS, MULTIPLES, AND PRIME WITH VIDEOS**

## HCF (HIGHEST COMMON FACTORS)

In mathematics, **the HCF (Highest Common Factor) of two or more integers is the largest positive integer that divides all of them without leaving a remainder**. For example, the **HCF of 24 and 36 is 12** because 12 is the largest positive integer that divides both 24 and 36 without leaving a remainder.

Finding the HCF of two or more integers can also be done using prime factorization. **To find the HCF of two or more integers using prime factorization, we take the smallest power of each prime factor that appears in all of the prime factorizations and multiply them together**.

For example, to **find the HCF of 12 and 20**, we first find their prime factorizations: 12 = 2^2 \times 3 and 20 = 2^2 \times 5. Then, we take the smallest power of each prime factor that appears in both factorizations and multiply them together:

The smallest power of 2 that appears is 2^2.

There is no common power of 3 that appears in both factorizations.

There is no common power of 5 that appears in both factorizations.

Therefore, the HCF of 12 and 20 is 2^2 = 4.

## FORMULA CONNECTING LCM AND HCF

There is a formula connecting the LCM and HCF of two integers. This formula is:

LCM(a, b) \times HCF(a, b) = a \times bIn other words, the product of the LCM and HCF of two integers is equal to the product of the two integers. This formula can be used to find either the LCM or HCF of two integers if the other one is known.

For example, if we know that the HCF of two integers a and b is 4, and their product is 240, we can use the formula to find their LCM as follows:

LCM(a, b) = \frac{(a \times b)}{HCF(a, b)} = \frac{240}{4} = 60Similarly, if we know that the LCM of two integers a and b is 60, and their product is 240, we can use the formula to find their HCF as follows:

HCF(a, b) = \frac{(a \times b)}{LCM(a, b)} = \frac{240}{60} = 4.

## CONCLUSION

In conclusion, LCM and HCF are important concepts in mathematics that are used to find the smallest positive integer that is divisible by all of a set of integers and the largest positive integer that divides all of a set of integers without leaving a remainder, respectively. Both LCM and HCF can be found using prime factorization, and there is a formula connecting them: LCM(a, b) \times HCF(a, b) = a \times b. This formula can be used to find either the LCM or HCF of two or more integers if the other one is known.

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