LCM of algebraic multiples: If the three numbers are 2a, 5a, and 7a (with a a positive integer), find their LCM in terms of a.

Introduction / Context:The Least Common Multiple (LCM) of numbers that are all multiples of a common factor can be computed by factoring that common part out and taking the LCM of the remaining coefficients. Here each term has a factor a, and the coefficients are small integers.

Given Data / Assumptions:

• Numbers: 2a, 5a, 7a
• a is a positive integer so that divisibility is well-defined.

Concept / Approach:LCM(k1*a, k2*a, k3*a) = a * LCM(k1, k2, k3) because each number contains a single factor a to the first power and the LCM keeps the highest power of each prime across the list. The coefficients 2, 5, and 7 are pairwise coprime, so their LCM is their product.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 65a, 75a: Coefficients 65 or 75 are not multiples of all of 2, 5, and 7.
• 70a^3: Exaggerates the power of a; the LCM includes a only to the first power here.

Common Pitfalls:

• Multiplying the a factor too many times rather than keeping its highest power across the numbers (which is 1 here).

Final Answer: