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: Factorise 2730 instead of guessing.
Break 2730 down into its prime factors:
$2730 = 2 \times 3 \times 5 \times 7 \times 13$

Step 2: Group the prime factors into three consecutive numbers.
We need to split these five primes into three factors that come out as consecutive whole numbers. Group $2 \times 7 = 14$, keep $13$ alone, and group $3 \times 5 = 15$. This gives the three numbers 13, 14, 15, which are indeed consecutive.

Step 3: Verify the product.
$13 \times 14 \times 15 = 13 \times 210 = 2730$
This confirms the grouping is correct, since it reproduces the given product exactly.

Step 4: Add the three numbers.
$13 + 14 + 15 = 42$

Step 5: Confirm no other grouping works.
Since 2730 has exactly the prime factors 2, 3, 5, 7 and 13, and 13 is itself prime, it can only appear as a whole factor on its own or combined with another prime while keeping the three numbers consecutive. The grouping above is the only one that lands on three consecutive integers, so the sum is uniquely 42.
\[ \boxed{42} \]