Select the odd number from the following alternatives based on whether it can be expressed as a product of three consecutive integers.

Introduction / Context:
This numerical reasoning question examines your ability to recognize hidden factorization patterns, specifically numbers that can be written as the product of three consecutive integers. Such patterns are useful in algebra, number theory, and competitive exam questions involving special forms of integers.

Given Data / Assumptions:

• The four given numbers are: 1716, 2730, 3360, and 4086.
• You must select the number that does not fit the common factorization pattern of the other three.
• Assume all numbers are positive integers.

Concept / Approach:
The approach is to test whether each number can be expressed in the form n * (n + 1) * (n + 2), where n, n + 1, and n + 2 are three consecutive integers. If three numbers share this property and one does not, that one is the odd term. This may require trial factorization or spotting familiar products.

Step-by-Step Solution:
Step 1: Factor 1716. One way is to check 11 * 12 * 13. Multiplying 11 * 12 gives 132, and 132 * 13 gives 1716, so 1716 equals 11 * 12 * 13, a product of three consecutive integers. Step 2: Consider 2730. Try 13 * 14 * 15. Multiplying 13 * 14 gives 182, and 182 * 15 gives 2730, so 2730 equals 13 * 14 * 15, which again is a product of three consecutive integers. Step 3: Check 3360. Try 14 * 15 * 16. Multiplying 14 * 15 gives 210, and 210 * 16 gives 3360, so 3360 equals 14 * 15 * 16, also a product of three consecutive integers. Step 4: Now test 4086. No simple choice of n gives n * (n + 1) * (n + 2) equal to 4086. If we try 15 * 16 * 17, we get 4080, which is close but not equal. Trying 16 * 17 * 18 gives a much larger value. Therefore, 4086 cannot be written as a product of three consecutive integers in the same neat way. Step 5: Since 1716, 2730, and 3360 all share this factorization property and 4086 does not, 4086 is the odd number.

Verification / Alternative check:
You can also factor each number into primes. For example, 1716 factors into 2 * 2 * 3 * 11 * 13, which can be regrouped as 11 * 12 * 13. Similarly, 2730 and 3360 admit groupings that correspond to 13 * 14 * 15 and 14 * 15 * 16. However, 4086 factors into 2 * 3 * 3 * 227 and does not rearrange into three consecutive integers. This prime factorization view confirms that 4086 breaks the pattern.

Why Other Options Are Wrong:

• 1716: Equal to 11 * 12 * 13, so it clearly fits the special product pattern.
• 2730: Equal to 13 * 14 * 15, so it also has the required product form.
• 3360: Equal to 14 * 15 * 16, so it conforms perfectly to the pattern of consecutive integer products.

Common Pitfalls:
Many students focus only on divisibility by small primes or number size and miss the specific product pattern. Another trap is to stop checking after spotting a partial pattern, such as divisibility by 3 or 5. The correct approach here is not just divisibility but matching the exact form n * (n + 1) * (n + 2). Ignoring the structure can lead to an incorrect choice.

Final Answer:
4086