To find a common denominator for the fractions \( \frac{12}{13} \) and \( \frac{13}{39} \), we first need to find the least common multiple (LCM) of the denominators 13 and 39.
The number 39 can be factored into prime factors: \[ 39 = 3 \times 13 \] The denominators are:
- 13
- 39
The LCM is the highest power of each prime factor appearing in the denominators:
- The prime factor 13 appears as \( 13^1 \) in both.
- The prime factor 3 appears as \( 3^1 \) in 39.
Thus, the LCM is: \[ \text{LCM}(13, 39) = 3^1 \times 13^1 = 3 \times 13 = 39. \]
Therefore, the common denominator for the fractions \( \frac{12}{13} \) and \( \frac{13}{39} \) is \( 39 \).
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