Question

Find the least common multiple.
67w2, 134
Write your answer as a constant times a product of single variables raised to exponents.

Answer 1

To find the least common multiple (LCM) of the two expressions \(67w^2\) and \(134\), we first need to factor each expression into its prime factors.

1. The first expression is:
\[
67w^2
\]
Here, \(67\) is a prime number, and \(w^2\) is a variable raised to an exponent.

2. The second expression is:
\[
134
\]
We can factor \(134\) as follows:
\[
134 = 2 \times 67
\]

Now we can summarize the factors:
- From \(67w^2\), we have the factors \(67\) and \(w^2\).
- From \(134\), we have the factors \(2\) and \(67\).

Next, we will find the LCM by taking the highest power of each prime factor appearing in the factorizations.

- For the prime \(2\): It appears in \(134\) as \(2^1\).
- For the prime \(67\): It appears as \(67^1\) in both \(67w^2\) and \(134\).
- For the variable \(w\): It appears as \(w^2\) in \(67w^2\).

Therefore, the LCM is obtained by taking:
\[
\text{LCM} = 2^1 \cdot 67^1 \cdot w^2
\]
This simplifies to:
\[
\text{LCM} = 2 \cdot 67 \cdot w^2
\]

Calculating \(2 \cdot 67\):
\[
2 \cdot 67 = 134
\]

Thus, the least common multiple can be expressed as:
\[
134w^2
\]

Finally, the LCM of \(67w^2\) and \(134\) written as a constant times a product of single variables raised to exponents is:
\[
\boxed{134w^2}
\]