What is the lcm of 16 and 28?

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 16 and 28. To solve it, we will find the multiples of both numbers and identify the smallest common multiple.

Answer

The least common multiple (LCM) of 16 and 28 is $112$.

Steps to Solve

Find the prime factorization of each number

Start by breaking down 16 and 28 into their prime factors:

For 16:
- $16 = 2 \times 8 = 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 = 2^4$
For 28:
- $28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7^1$

So, we have:

$16 = 2^4$
$28 = 2^2 \times 7^1$

Identify the highest powers of all prime factors

Next, we take the highest power of each prime number from the factorizations:

For the prime number 2: the highest power is $2^4$ (from 16).
For the prime number 7: the highest power is $7^1$ (from 28).

Multiply the highest prime powers together

Now, we multiply these highest powers to find the LCM:

$$ \text{LCM} = 2^4 \times 7^1 $$

Calculating this we get: $$ \text{LCM} = 16 \times 7 = 112 $$

The least common multiple (LCM) of 16 and 28 is $112$.

More Information

The least common multiple is useful in various applications, including solving problems that involve finding common denominators in fractions, scheduling, and understanding patterns in multiples. The LCM of two numbers is effectively the smallest number that both original numbers can evenly divide into.

Tips

One common mistake is to assume the LCM can be found by simply multiplying the two numbers together, which is incorrect unless the numbers are coprime (having no common factors other than 1).
Another mistake is to incorrectly find the prime factorization. It's essential to ensure the factorization is accurate before proceeding to find the LCM.

Thank you for voting!