The smallest fraction, which each of 6/7, 5/14, 10/21 will divide exactly is

### Concept & Formula When a question asks for the smallest number or fraction *which is exactly divided by* other fractions, it is asking for the Least Common Multiple (LCM) of those fractions. The formula for finding the LCM of fractions is: $$\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$ ### Step-by-Step Solution **Given fractions:** * $\frac{6}{7}, \frac{5}{14}, \frac{10}{21}$ **Calculation:** * Step 1: Identify the numerators and denominators. * Numerators: $6, 5, 10$ * Denominators: $7, 14, 21$ * Step 2: Find the LCM of the numerators ($6, 5, 10$). * Multiples of $10$: $10, 20, 30...$ * $30$ is perfectly divisible by both $6$ and $5$. * $\text{LCM}(6, 5, 10) = 30$ * Step 3: Find the HCF of the denominators ($7, 14, 21$). * All numbers are multiples of $7$. * $\text{HCF}(7, 14, 21) = 7$ * Step 4: Substitute the values back into the fraction formula. $$\text{LCM} = \frac{30}{7}$$ ### Exam Strategy & Shortcut Memorize the fraction rules perfectly: "LCM = LCM of top / HCF of bottom". The moment you look at the numerators ($6, 5, 10$), your mind should instantly jump to $30$. Looking at the denominators ($7, 14, 21$), $7$ stands out as the highest common factor. This gives you $\frac{30}{7}$ within 5 seconds, allowing you to instantly select option (a). ### Common Pitfall A common point of confusion is mixing up the fraction rules and calculating $\frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$ instead. That formula finds the *greatest common divisor* (HCF) of fractions, not the smallest common multiple (LCM). ### Final Answer **Therefore, the correct answer is 30/7.**