lcm of 30 and 16

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 30 and 16. To solve it, we will identify the multiples of each number and find the smallest multiple common to both.

Answer

The least common multiple of 30 and 16 is $ 240 $.

Steps to Solve

Find the prime factorization of each number

Start by finding the prime factors of 30 and 16.

The prime factorization of 30 is: $$ 30 = 2^1 \times 3^1 \times 5^1 $$
The prime factorization of 16 is: $$ 16 = 2^4 $$

List the highest powers of each prime factor

Next, identify the highest powers of all the prime factors present in either number.

For prime number 2, the highest power is $2^4$ (from 16).
For prime number 3, the highest power is $3^1$ (from 30).
For prime number 5, the highest power is $5^1$ (from 30).

Multiply these highest powers together

Now, multiply the highest powers of each prime factor together to find the LCM.

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

Calculate the product

Now, compute the product to find the LCM.

First, calculate $2^4 = 16$.

Then calculate: $$ 16 \times 3 = 48 $$ $$ 48 \times 5 = 240 $$

Final Calculation

Thus, the least common multiple of 30 and 16 is: $$ \text{LCM} = 240 $$

The least common multiple (LCM) of 30 and 16 is ( \text{LCM} = 240 ).

More Information

The LCM is useful in problems involving addition or subtraction of fractions with different denominators, helping to find a common denominator for easier computation.

Tips

A common mistake is to simply list multiples and find the smallest one, which can be time-consuming for larger numbers. It's more efficient to use prime factorization.
Another mistake is forgetting to consider all prime factors; ensure to include the highest power of each prime factor.

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