If three numbers are $2a$, $5a$ and $7a$, what will be their LCM?

### Concept & Formula The Least Common Multiple (LCM) of algebraic terms is found by multiplying the LCM of their numerical coefficients by the highest power of each variable present. $$\text{LCM}(cx, cy) = \text{LCM}(c) \times \text{Highest Power of Variables}$$ ### Step-by-Step Solution **Given:** * The three numbers are $2a$, $5a$, and $7a$. **Calculation:** 1. Separate the numerical coefficients from the variables: * Coefficients: 2, 5, 7 * Variable part: $a, a, a$ 2. Find the LCM of the numerical coefficients: * Since 2, 5, and 7 are all prime numbers, their LCM is simply their product. * $\text{LCM}(2, 5, 7) = 2 \times 5 \times 7 = 70$ 3. Find the LCM of the variable parts: * The highest power of the variable '$a$' across all three terms is just $a^1$ (or $a$). 4. Combine both parts: * $\text{Total LCM} = 70 \times a = 70a$ ### Exam Strategy & Shortcut Recognize immediately that 2, 5, and 7 share no common factors (they are mutually prime). Therefore, you just multiply them ($2 \times 5 \times 7 = 70$). Because the variable '$a$' is common to all terms and raised to the first power in each, it simply tags along once. $70 \times a = 70a$. ### Common Pitfall A very common mistake in algebra is multiplying the variables together as if they were independent distinct terms, resulting in $70a^3$. Remember, LCM looks for the *highest power* of the common base present in any single term, it does not multiply the bases together. ### Final Answer **Therefore, the correct answer is $70a$.**