The sum of three consecutive multiples of $3$ is $72$. What is the largest number?

### Concept & Strategy Any set of consecutive multiples (like multiples of $3$) forms an Arithmetic Progression (AP). The most powerful property of an AP with an odd number of terms is that its average strictly equals its middle term. $$\text{Middle Term} = \frac{\text{Sum}}{\text{Number of Terms}}$$ ### Step-by-Step Solution * **Given:** The sum of $3$ consecutive multiples of $3$ is $72$. * Calculate the average to find the middle term directly: $$\text{Middle Term} = \frac{72}{3} = 24$$ * Since the numbers are consecutive multiples of $3$, they are spaced by $3$. * The middle number is $24$. * The sequence is: $(24 - 3), \textbf{24}, (24 + 3)$. * The sequence is: $21, 24, 27$. * The question asks for the largest number, which is $27$. ### Exam Strategy & Shortcut Do not use the traditional algebra method of setting up $x + (x+3) + (x+6) = 72$. While correct, it forces you to solve $3x + 9 = 72 \Rightarrow 3x = 63 \Rightarrow x = 21$, and then calculate $21 + 6 = 27$. Using the average property ($72 / 3 = 24$, then $24 + 3 = 27$) requires zero scratch paper and solves the problem in under $5$ seconds mentally. ### Common Pitfall A standard trap is identifying the first number ($21$) or the middle number ($24$) and accidentally choosing it from the options. Always box or underline the specific condition requested ("largest number") before glancing at the choices. ### Final Answer **Therefore, the correct answer is 27.**