To find the least common denominator (LCD) of the fractions \( \frac{3k}{40} \) and \( \frac{k}{18} \), we need to find the least common multiple (LCM) of the denominators 40 and 18.
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Finding the prime factorization of each denominator:
- \( 40 = 2^3 \times 5^1 \)
- \( 18 = 2^1 \times 3^2 \)
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Finding the LCM: To find the LCM, we take the highest power of each prime factor that appears in either factorization.
- For \( 2 \): the highest power is \( 2^3 \) (from 40).
- For \( 3 \): the highest power is \( 3^2 \) (from 18).
- For \( 5 \): the highest power is \( 5^1 \) (from 40).
Therefore, the LCM is: \[ LCM = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]
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Calculating the LCM: \[ 8 \times 9 = 72 \] \[ 72 \times 5 = 360 \]
Thus, the least common denominator (LCD) of \( \frac{3k}{40} \) and \( \frac{k}{18} \) is \( 360 \).
Therefore, the answer is \( 360 \).