If n is a natural number, then the expression (6n² + 6n) is always divisible by which of the following?

Problem Restatement

Find the highest guaranteed divisor among the choices for 6n² + 6n when n is any natural number.

Given / Factorization

6n² + 6n = 6n(n + 1)

Concept / Approach

• Among two consecutive integers n and (n + 1), one is even.
• Therefore n(n + 1) is even → divisible by 2.
• Since there is an explicit factor 6 = 2 × 3, the whole product has at least 2 × 2 × 3 = 12.

Step-by-Step Reasoning

6n(n + 1) = (2 × 3) × n(n + 1)n(n + 1) contributes at least one factor 2Hence overall divisibility includes 2 × 2 × 3 = 12

Common Pitfalls

• Marking 6 as the answer; while true, 12 is also always a divisor and is larger.
• Assuming divisibility by 18; not guaranteed because an extra factor of 3 is not assured for all n.

Final Answer

12