Give some LCM and HCF.

Understand the Problem

The question is asking for examples of Least Common Multiples (LCM) and Highest Common Factors (HCF), which are concepts used in mathematics to find common multiples and factors of numbers.

Answer

LCM(12, 15) = $60$, HCF(12, 15) = $3$.

Steps to Solve

Find the LCM of two numbers

To find the Least Common Multiple (LCM) of two numbers, you can use the prime factorization method. First, factor each number into its prime components.

For example, let's find LCM(12, 15):

12 = $2^2 \cdot 3^1$
15 = $3^1 \cdot 5^1$

Next, determine the highest power of each prime that appears in either factorization:

For 2: $2^2$
For 3: $3^1$
For 5: $5^1$

Then, multiply these together:

$$ \text{LCM} = 2^2 \cdot 3^1 \cdot 5^1 = 4 \cdot 3 \cdot 5 $$

Calculate the product to find the LCM

Now, calculate the product:

$$ \text{LCM} = 4 \cdot 3 \cdot 5 = 60 $$

So, LCM(12, 15) = 60.

Find the HCF of two numbers

To find the Highest Common Factor (HCF), we'll use the same numbers, 12 and 15, and identify the common prime factors.

Looking at our previous factorization:

12 = $2^2 \cdot 3^1$
15 = $3^1 \cdot 5^1$

The only common prime factor is 3. The minimum power of this factor is:

For 3: $3^1$

Thus, the HCF is:

$$ \text{HCF} = 3 $$

The Least Common Multiple (LCM) of 12 and 15 is $60$, and the Highest Common Factor (HCF) is $3$.

More Information

The LCM is a useful concept for finding common denominators when adding fractions, while the HCF helps in simplifying fractions. Understanding these concepts allows for easier manipulation of numbers in various mathematical contexts.

Tips

Mistaking LCM for HCF: Confusing the definitions of LCM and HCF is common. Remember, LCM deals with multiples (product) while HCF deals with factors (divisibility).
Incorrect prime factorization: Ensuring all numbers are correctly factored into primes is crucial.

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