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.
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The first expression is: \[ 67w^2 \] Here, \(67\) is a prime number, and \(w^2\) is a variable raised to an exponent.
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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: \[ \boxed{134w^2} \]
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