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Consecutive integers are numbers that follow one another in order, without any gaps. These can be represented as , n+1, and n+2 if we are looking at three consecutive numbers. For example, examples of three consecutive integers include 1, 2, 3 or 10, 11, 12.

Understanding consecutive integers is important for solving various mathematical problems. They often appear in problems dealing with divisibility, sequences, and patterns. When working with consecutive integers, you are essentially dealing with a sequence, where each number is exactly one more than the previous number. This predictable pattern makes it easier to apply different mathematical rules and properties.

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division. Divisibility rules for 2 and 3 are particularly straightforward and helpful.

The rule for divisibility by 2 states that a number is divisible by 2 if it is even. This means it ends in 0, 2, 4, 6, or 8. The rule for divisibility by 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.

Combining these rules, in the context of our problem on consecutive integers, simplifies the process of verifying divisibility by 6. Since 6 is the product of 2 and 3, if you can show that one number is even, and another number in the sequence is divisible by 3, you prove that their product is divisible by 6.

A mathematical proof is a logical argument that demonstrates the truth of a given statement. It uses previously established facts, definitions, and theorems to show that a conclusion follows necessarily from assumptions.

In our exercise, we are asked to prove that the product of any three consecutive integers is divisible by 6. Here's a simple breakdown of the steps involved in this proof:

• Represent the three consecutive integers as n, n+1, and n+2.
• Identify divisibility by 2: At least one of these integers must be even, and therefore divisible by 2.
• Identify divisibility by 3: Among the three integers, one must be divisible by 3.
• Combine these facts: Since one number is divisible by 2 and another by 3, their product must contain the factors 2 and 3. Consequently, it is divisible by 6.

This methodical approach ensures your conclusions are based on solid reasoning and established rules, providing a reliable proof of the statement.