Question

The product of three consecutive numbers is 2730. What is the sum of the three numbers?

• A. 39
• B. 42
• C. 45
• D. None of these

Hint:

The product of three numbers close to 14 is 2730, so test 13 times 14 times 15 directly, or factor 2730 into primes and group them into three consecutive integers.

Answer 1

The Correct Option is B

Step 1: Set up the equation.
Let the three consecutive numbers be \(n\), \(n+1\) and \(n+2\). We are told their product is 2730, so:
\[ n(n+1)(n+2) = 2730 \]

Step 2: Estimate the size of \(n\).
Since the three numbers are close together, their product is roughly \(n^3\). Taking the cube root of 2730 gives a value near 14, so we should look for \(n\) somewhere around 13 or 14.

Step 3: Test nearby values.
Try \(n=13\): \(13 \times 14 \times 15\). First, \(13 \times 14 = 182\). Then \(182 \times 15 = 2730\). This matches exactly.

Step 4: Find the sum.
The three consecutive numbers are 13, 14 and 15, so their sum is:
\[ 13 + 14 + 15 = 42 \]

Step 5: Check the other options.
A sum of 39 or 45 would come from a different set of three consecutive numbers, such as 12+13+14=39 or 14+15+16=45, and neither of those products equals 2730, since \(12 \times 13 \times 14 = 2184\) and \(14 \times 15 \times 16 = 3360\). Since 42 does correspond to a set whose product is exactly 2730, None of these does not apply.

Final Answer:
The three consecutive numbers are 13, 14 and 15, and their sum is 42.
\[ \boxed{42} \]