1. **State the problem:** Find the least common multiple (LCM) of the fractions $\frac{3}{7}$, $\frac{5}{5}$, and $\frac{1}{6}$.\n\n2. **Recall the formula for LCM of fractions:** The LCM of fractions is given by $$\text{LCM}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{\text{LCM}(a,c,e)}{\text{GCD}(b,d,f)}$$ where LCM is the least common multiple and GCD is the greatest common divisor.\n\n3. **Identify numerators and denominators:** Numerators are 3, 5, and 1. Denominators are 7, 5, and 6.\n\n4. **Calculate LCM of numerators:**\n- Prime factors: 3 = 3, 5 = 5, 1 = 1\n- LCM(3,5,1) = 15 (since 3 and 5 are prime and 1 does not affect LCM)\n\n5. **Calculate GCD of denominators:**\n- Denominators: 7, 5, 6\n- Factors: 7 (prime), 5 (prime), 6 = 2 \times 3\n- No common prime factors, so GCD(7,5,6) = 1\n\n6. **Apply the formula:**\n$$\text{LCM of fractions} = \frac{15}{1} = 15$$\n\n**Final answer:** The least common multiple of the fractions $\frac{3}{7}$, $\frac{5}{5}$, and $\frac{1}{6}$ is $15$.