LCM or GCD?

Let the smallest fraction be $\dfrac ab$ , where $a$ and $b$ are positive coprime integers. Then, we have $\dfrac ab \div \dfrac 6{35} = \dfrac ab \times \dfrac {35}6 = n_1$ , $\dfrac ab \times \dfrac {21}{10} = n_2$ , and $\dfrac ab \times \dfrac {49}{15} = n_3$ , where $n_1$ , $n_2$ , and $n_3$ are integers. For $\dfrac ab$ to be the smallest, $a$ should be the smallest, while $b$ the biggest. For $n_1$ , $n_2$ , and $n_3$ to be integers, $\dfrac a6$ , $\dfrac a{10}$ , and $\dfrac a{15}$ must be integers and the smallest $a$ is $a_{min} = \text{lcm } (6,10,15) = 30$ . Similarly, $\dfrac {35}b$ , $\dfrac {21}b$ , and $\dfrac {49}b$ must also be integers and the biggest $b$ is $b_{max} = \gcd (35,21,49) = 7$ . Therefore, $\dfrac ab = \dfrac {30}7 \approx \boxed{4.28}$ .