If r, s, and t are consecutive positive positive multiples of 3, is rst divisible by 27, 54, or both?

Answer

To determine whether the product \( rst \) of three consecutive positive multiples of 3 is divisible by 27, 54, or both, we can follow these steps:

### Step 1: Define the Variables
Let \( r, s, t \) be the consecutive positive multiples of 3. We can express them in terms of a positive integer \( x \):
- \( r = 3x \)
- \( s = 3(x + 1) = 3x + 3 \)
- \( t = 3(x + 2) = 3x + 6 \)

### Step 2: Calculate the Product
Now, we calculate the product \( rst \):
\[ rst = r \times s \times t = (3x) \times (3x + 3) \times (3x + 6) \]
This can be simplified as:
\[ rst = 3x \times 3(x + 1) \times 3(x + 2) = 27x(x + 1)(x + 2) \]

### Step 3: Analyze the Divisibility
Now we need to check if \( rst \) is divisible by 27, 54, or both:
- We already see that \( rst \) contains \( 27 \) as a factor.
- Next, we need to check if \( x(x + 1)(x + 2) \) is divisible by 2.

### Step 4: Check for Evenness
The expression \( x(x + 1)(x + 2) \) represents the product of three consecutive integers. Among any three consecutive integers, at least one of them is even. Therefore, \( x(x + 1)(x + 2) \) is guaranteed to be divisible by 2.

### Step 5: Conclusion on Divisibility
Since \( rst = 27 \times x(x + 1)(x + 2) \) is divisible by 27 and \( x(x + 1)(x + 2) \) is divisible by 2, we can conclude:
\[ rst \text{ is divisible by } 27 \times 2 = 54. \]

Thus, \( rst \) is divisible by both 27 and 54.

### Final Answer
The product \( rst \) is divisible by both 27 and 54.

---